The terms "Tangent" and "Reality" are different than a "Plane", and different than the "Multiverse" (an entire set of Planes of Existence).
Tangents:
The word "Tangent" is from Alternity...it refers to alternate timelines and parallel worlds. Could also just call it a "Timeline". Perhaps different terminology could be gleaned from the AD&D Chronomancer supplement. Tangents are also like the Dimension X campaign model from d20 Future.
I'd use the in-story word "Tangent" for alternate timelines of the D&D Multiverse and its worlds. If it's just a minor discrepancy or contradiction (of which there are many in D&D history, since the modules, sourcebooks, novels, and videogames were written by many different authors), both of the contradictions are "Minor Tangents". For example, if one source says an NPC was born on one date, and another says another date, both are true...they're just two slightly different timelines...Minor Tangents.
Major Tangents are timelines which continue to diverge in significant ways.
Tangents are different versions of a recognizable D&D World. Or they could be a mix of two or more D&D Worlds.
The published "WotC Multiverse" contains all of the published Tangents. There is a main timeline...the Mainline Tangent. But all of the published in-story contradictions and discrepancies are also part of the "WotC Multiverse"...they're just a different Tangent within that Multiverse. They are all "canon" and all "official"...they all "happened".
Some examples of Major Tangents:
The 4e timeline reboot of Dark Sun is apparently a different timeline than the previously published 2e/3e timeline.
The 5e reboot of Ravenloft via the Curse of Strahd is different timeline than the 2e/3e RL timeline. CoS includes iconic characters from various times, and is set in a time which contradicts the previous timeline. So there's a "Curse of Strahd Tangent" (which is the Tangent which is currently supported in 5e), and a "Classic Ravenloft Tangent" which follows the rest of the previously published books. (Plus many Minor Tangents for each of the minor timeline snags in older books.)
In Mystara, the events in the module Master of Desert Nomads were officially placed in two different times (1200 AC or ~1005/1006 AC). In one Tangent, that war happened between 1000 AC and 1010 AC, and in the other Tangent, in 1200 AC.
Tangents are different than "Worlds" because each Tangent exists in a different Material Plane (and a different Multiverse, to the extent that a Tangent involves extraplanar events or characters). Each Tangent is a slightly different time-strand within the Material Plane or Multiverse. In fact, each Tangent is a different version of the D&D Multiverse. Because even extraplanar locales and deities can exist in multiple timelines.
Even a slight Minor Tangent spawns an entire D&D Multiverse. Why? Because when a being (such as a deity) who lives in an Outer Plane scries an event the World, they only see one particular event...they don't see all of the possible variations of events. So even deities and the outer planes must be encompassed within a particular Tangent. However, there are means by which (for story purposes) a deitiy (or mortal) can access another Tangent...but it's not a usual occurrence.
A Tangent can't be reached by skyship or spelljammer (unless of course, a special event pulls the ship out of the main Tangent). It's like in Star Trek...you can't just pilot a course to the Mirror Dimension, or to an alternate timeline. There has to be some sort of chronomantic/planar event or rift which brings the party into another Tangent.
Tangents can only be reached by DMs discretion...but there are rules in the 2e Chronomancer (time-travel) supplement which are relevant, and are presumably still accessible in 5e. There are also methods of reaching other Tangents described in the Dimension X campaign model of d20 Future and in the Tangents sourcebook of the Alternity RPG. Yes, from a planar pespective, Tangents could be looked open as "planes" or "worlds within the Material Plane", and they are practically the same as planes...except that they haven't (thus far) been consciouslly mapped as part of the "Planes of Existence" (otherwise they'd be many different Torils, and any different Sigils on the map)...and a particular Tangent could also have its own Multiverse. And so though Tangents could be reached through planar travel, via the means for traveling to "Alternate Primes" mentioned in the 1e Manual of the Planes, and their equivalent in later planar manuals, they are a special case.
Realities:
Various rules systems have been used as lenses to view the D&D Worlds (the settings). These different rules systems (or "game universes") result in real differences in the same setting. For example, there were no more 1e-style Assassins or Barbarians after the Time of Troubles; and in Mystara, when it switched from BECMI to 2E, we discovered that some Fighter NPCs were really Rangers and Paladins, and some Thieves were really Bards. That's a pretty big difference. Canonically, these different "game universes" are different Realities, which are a higher "level" than the Multiverse.
An official in-game word for "rules system" or "game universe" is "Reality". This is from the Dragon magazine article "Up, Away, and Beyond" by Bruce Heard, then D&D Brand Manager. This article describes how the Classic D&D/BECMI rules system is really a different Realilty from the AD&D rules system, but that CD&D Immortals can actually notice an in-story difference when they travel to the AD&D Reality and are translated into AD&D Deities. Here's the text which canonically states that BECMI Mystara is in a different reality from the 2E worlds:
"Where are the AD&D(R) game worlds?"
"Before tackling the connection with AD&D 2nd Edition game, it would be useful to clarify the existence of the D&D game world in relation to the AD&D game universe. The various AD&D game worlds occupy physical positions in space. These worlds can be reached by spelljammer ships. The D&D world and its universe /do not exist at all/ in the AD&D world, which explains why they were not mentioned in the SPELLJAMMER(TM) boxed set. The two universes cannot be connected by normal space navigation."
"To travel from one game universe to another, the characters must invoke a /reality shift/. This is different from opening a magical /gate/ or casting a /wish/. The /reality shift/ causes the character to travel not only across space and parallel dimensions, but also across the very fabric of reality, which causes the characters' transformation. The characters would not actually be aware of this alteration. Incompatible aspects of the character's cultural background, personal history, and memory are instantly and painlessly modified to fit the new world."
"Characters returning from a trip across a /reality shift/ would instantly revert to their former selves. No experience is lost in the process, since the party would be "persuaded" that all of their adventures took place in whatever world they occupy! They simply cannot suspect or even grasp the concept that there are effectively two or more different realities."
"For simplicity's sake, we'll establish that time flows in the same direction on both sides of the /reality shift/. A character spending 10 years in one reality before returning to his previous reality would reappear 10 years later and 10 years older as well."
"/Reality shifts/ may be effected only by greater gods in the AD&D game or the most powerful Immortals of the D&D game. These deities would first have to discover the another universe /that logically does not exist/--something difficult to achieve even for them. Some of those divine beings crossed over and were subjected to the unavoidable transformation. [Webmaster's note: for example, Thor and Orcus.] Although unaware of their personal metamorphosis, divine beings retain their memories about their previous worlds, unlike mortals."
"When a deity leaves for another reality, an alter ego of the deity is created. The alter ego is a metaphysical force whose usefulness lies in its ability to grant spells to worthy clerics in the deity's absence. When the deity returns, the alter ego dissipates (or reappears in the other reality, if the deity gained worshippers there)."
"Unless they are being worshiped in several realities, divine beings tend to remain in their home realities. They experience discomfort caused by the strange alteration of their senses when crossing realities, and they dislike the vague, unexplainable discrepancies between the two universes' realities. Even deities fear the unknown."
I offer further, straightforward elaborations based on Heard's article:
Each D&D rules iteration is a different Reality.
Each D&D World (setting) has been viewed in various Realities (rules iterations) in various official TSR/WotC publications.
There are real in-story differences between different Realities of the same World. For example, in the CD&D Reality, Mystara exists in the Gold Box/WotI cosmology, but in the AD&D2e Reality, Mystara is another Prime plane within the Great Wheel.
The various combinations of Worlds and Realities continue to exist "off screen" even when TSR/WotC ceased publishing that combination. For example, even though CD&D Mystara was no longer published after 2e Mystara came out, CD&D Mystara continues to exist. Despite TSR's policy of "later trumps earlier", the vast library of CD&D Mystara books weren't just "disappeared" by the few 2e Mystara products that came out before the world was cancelled. CD&D Mystara just went "off screen" and the 2e Reality of Mystara (which had existed from the beginning of time in the Great Wheel) walked "on screen" in official publications.
All Realities extend forever into the past and future of each World. For example, the Second Edition Reality of Toril extends throughout its past and future, even though, from an OOG perspective, the Time of Troubles was designed to begin that Reality. This is evident by the fact that the Arcane Age products use the 2e rules, even though they happened before the ToT. Likewise, after the SAGA System ceased production, the Dragonlance Chronicles-era and Fifth Age-era appeared in 2e and 3e rules, even though they "should" have used the 1e and SAGA rules.
However, those worlds such as Forgotten Realms, Dragonlance, and Greyhawk, which have received in-story support for rules-shifts, also exist in the "Legacy Reality" (aka Edition Shift Reality), wherein the "expected" rules systems are used in the order they shifted through their publication history. For example, in the Legacy Reality, the FR Arcane Age would only be depicted with 1e rules. And in the Legacy Reality, the Fifth Age of Krynn is only depicted with SAGA System, not with the later 3e versions of Fifth Age spellcasting classes. This Legacy Reality exists parallel to the pure "First Edition Reality", "Second Edition Reality", etc. This Legacy Reality was the main timeline, which was replaced by the Fifth Reality.
Rules updates within a single iteration, such as 3.0e to 3.5e, could be called a "Reality Shimmer". It's not a different Reality, but a rules update shimmers throughout the setting, subtly changing certain in-story effects. A "Reality Shimmer" is to the rules system what "Timeline Eddy" (a continuity discrepancy) is to the setting. In D&D Next, there ought to recognition for all the Realities and alternate Timelines.
Each Reality keeps the essential in-story features of other Realities of the same World while respecting and sifting out the inherent differences in cosmology, classes, and so forth. For example, the in-story tidbits about Bargle that were included in the 4e web enhancement (which is a glimpse into the 4e Reality of Mystara) would be relevant to CD&D Mystara. Also, in the First Edition Reality of Forgotten Realms, the Time of Troubles and Spellplague still happened, but the in-story effects are interpreted in a way that keeps the 1e rules system.
All the D&D Worlds exist "off stage" in all the rules Realities. For example, there are BECMI versions of Toril, Krynn, and Eberron existing in the Classic Reality which is usually associated with Mystara. And 1e versions of Dark Sun, Jakandor, and Ghostwalk. Original D&D versions of all the Worlds, with Fighting-Men instead of Fighters.
The various other TSR/WotC rpgs (such as Alternity and Amazing Engine) are each a separate Reality. The concept is applied also to different non-rpg media through which the worlds are depicted, such as boardgame rules, Endless Quest gamebook rules, etc.
Extending this beyond official TSR/WotC publications: each DM's set of house-rules are a different Reality, even if the group plays "by the book."
The main Realities of the D&D Worlds:
The Original D&D Reality
Chainmail miniatures reality shimmer
The Classic D&D Reality
B/X reality shimmer
BECMI reality shimmer
War Machine reality shimmer (a temporary shimmer, when the game-play shifts to mass combat rules)
Rules Cyclopedia/Wrath of the Immortals reality shimmer (with entirely different Immortals-level rules)
The AD&D First Edition Reality
1.5e reality shimmer (Unearthed Arcana class re-arrangement, Comeliness stat, non-weapon proficiencies, etc.)
Battlesystem reality shimmer (a temporary shimmer, when the game-play shifts to mass combat rules)
The AD&D Second Edition Reality
2.5e reality shimmer (Skills & Powers)
2e Battlesystem reality shimmer (a temporary shimmer, when the game-play shifts to mass combat rules)
The SAGA System Reality
The D&D Third Edition Reality
3.5e reality shimmer
Chainmail d20 reality shimmer (features the 3e logo, but is a different game)
The D&D Fourth Edition Reality
Essentials shimmer
The D&D Fifth Edition Reality
5E Basic Rules shimmer
The Legacy (aka Edition-Shift) Reality is a Reality which uses the "proper" rules lenses for different eras of history, such as 1e for the Arcane Age of Forgotten Realms, even though those products actually featured 2e rules). Not all of the settings had in-world catastrophes to mark the editions shifts within the Legacy Reality; for example Mystara's Poor Wizard's Almanacs were in CD&D; then the next year they were in AD&D2e, with no commentary or earth-shattering event. Our lens simply shifted from the Classic Reality to the 2E Reality. Yet in the Legacy Reality, all of the worlds definitely went through in-story events which are equivalent in magnitude to the Time of Troubles, Spellplague, and Second Sundering.
In-story terms for the Realities: (Though only deities know of, and can the perceive, the difference between Realities):
The Original Reality (OD&D)
The Classic Reality (B/X D&D, BECMI D&D, Black Box/Rules Compendium/Wrath of the Immortals D&D)
The First Reality (AD&D 1e, Unearthed Arcana)
The Second Reality (AD&D 2e)
The Saga Reality (Saga System)
The Third Reality (D&D 3.0e, D&D 3.5e)
The Fourth Reality (D&D 4e, 4e Essentials)
The Fifth Reality (D&D 5e)
The Legacy Reality (a timeline where all the D&D worlds actually metamorphosed into the various Realities in a synchonized way, such as via the events of Die Vecna Die.)
Other realities:
The Adventure System Cooperative Play board game Reality
Spellfire trading card game Reality
The Great Khan Game Reality. If more of Faerun were to be seen through this lense, then other lands would be featured besides the Whamite Isles.
Orcwars! boardgame Reality. A depiction of the Broken Lands of Mystara.
The DragonQuest Reality
The Gamebook Reality
The Endless Quest gamebook reality shimmer
The Super Endless Quest/AD&D Adventure Gamebook reality shimmer.
The Catacombs Solo Quest reality shimmer
The 1 on 1 Adventure Gamebooks reality shimmer
The Amazing Engine Reality
The Alternity Reality
The d20 Modern Reality (closely related to the 3e Reality)
The Realities of non-D&D TSR/WotC rpgs of a similar era might be notionally "related", but are still different realities, as evinced by the Alternate World Gate description (below).
OD&D-era Realities: Boot Hill First Edition Reality, Metamorphosis Alpha Reality
1e-era RPG Realities, such as Marvel Super Heroes RPG Reality, Boot Hill Second Edition Reality, Indiana Jones RPG Reality, Star Frontiers RPG Reality
2e-era RPGs, such as Buck Rogers RPG Reality and Boot Hill Third Edition Reality
3e/d20-era Realities, such as Wheel of Time d20 Reality, Star Wars d20 Reality, etc.
Realities of licensed games which WotC no longer supports (such as Star Wars and Marvel Super Heroes) would probably not be mentioned, but their place in the scheme could be inferred.
Various D&D video game realities. To the extent they operate on different rules than the pen & paper rules, they are different Realities.
Each medium of audio-visual depiction of the D&D Worlds could be considered its own Reality:
The Movie Reality. This Reality is the World of Izmer (D&D films #1 and #2), Nerath (D&D film #3; the Book of Vile Darkness), and also the Dragonlance animated film.
The Television Reality. So far, only the Realm of the D&D Cartoon Show.
The Comics Reality
Likewise for the novels set in D&D Worlds:
The Novels Reality
AC4: The Book of Marvelous Magic also speaks of these "game worlds". Here's the text:
ALTERNATE WORLD GATE
These are various odd, unfamiliar items. When touched, each item/gate summons a being from an alternate world who appears within one turn. At that time, a door appears near the item (even if in midair), and a being steps through the door, grabs the item before the characters can react (even if it is apparently secured, stored, or held), and steps back through the doorway. If the characters react quickly you may allow them to step through the door and arrive in the alternate world. Once in such a world, the characters must locate a gem of returning (q.v.) to return to the D&D® game world. The characters might not remain human after passing through the gate, depending on the alternate world they visit. You and the players should discuss the change at that time; severe changes should be optional, not forced. Alternate world gates should be selected, not randomly determined. The various items and beings linked to them are:
Blackjack: This small, heavy item is made of leather wrapped about a strip of steel. It summons an agent from the TOP SECRET® game: a short, wiry, blond human wearing a black woolen outfit. The agent might stop and eye the characters suspiciously but does not speak.
Laser Pistol: This is a plastic and chrome cylinder with a grip on one end. It cannot be fired without the user first spending 2-5 turns experimenting. The pistol summons a security robot from the GAMMA WORLD® game: a metallic humanoid with two arms and two tentacles, each about 33 inches long. The robot may mutter something about "restricted areas" but will not attack or otherwise converse.
Lute: This musical instrument is 2 feet long. A long, thin neck makes up one half and a round, flat-topped ornate box the other. Strings of unknown material are attached to each end. The lute summons a bard from the ADVANCED DUNGEONS & DRAGONS® game—a normal wellarmed but unarmored human clad in green and carrying a flute. He may mutter something about incompatibility, but will not otherwise converse.
Medal: This is a small blue ribbon with a pin on the back and adorned with a silver metal object of odd design. The medal summons an ace pilot from the DAWN PATROL® game—a normal man garbed in strange clothes who wears a leather helmet, a strange device over the eyes, and a long scarf. Though silent, he might, if encountered outdoors, examine the sky carefully before departing.
Pocket Tool: This odd device is 3 inches long and may be unfolded to reveal a knife, corkscrew, and various other utensils of fine metal manufacture. The tool summons a Yazirian from the STAR FRONTIERS® game—a man-sized monkeylike being with membranes between its arms and body. The creature may bare its teeth and snarl at the characters but will not otherwise converse.
Star: This silver five-pointed item is apparently a brooch. The star summons a sheriff from the BOOT HILL® game—a normal man clad in fine but thin black leather who wears a metallic device strapped to each hip. He pins the star to his vest and might draw one of his hip devices, twirl it with one finger, and replace it. He then winks solemnly at the characters and departs.
Violin Case: This strangely-shaped box has three hasps; if opened, a golden, furry lining is seen but the box is empty. Both the lining and the material of the box are unfamiliar. The case summons a thug from the GANGBUSTERS™ game—a human clad entirely in black, wearing a cloth hat, and carrying an odd-looking metal tube with two handles and a large rounded middle. He may wave the device about threateningly but will not attack and cannot speak intelligently.
Note: This text from AC4 affirms that, since each of those games used different rules, they are differerent Realities. The later iterations exist in other Realities or reality shimmers, but share some or all of the in-game fluff. For example Gamma World 4E is in the 4E Reality. And Star Frontiers exists in the d20 Modern Reality as the Star Law campaign model.
The bard with the lute is a cross-over from the 1E Reality. The bard is probably from Oerth, since that was the "core setting" of the 1E era.
Making sense of catastrophic rules shifts:
TSR and WotC has used in-story events to justify shifts from one rules system to another. Please note that I'm not saying that it's a good thing that they did this, I'm simply saying that they did do it—these in-story rules shifts are a part of "official" depictions of those settings. Examples include:
Time of Troubles (1e Abeir-Toril to 2e Abeir-Toril)
Fate of Istus (1e Oerth to 2e Oerth)
Wrath of the Immortals (BECMI Mystara to Rules Cyclopedia/WotI Mystara, with changed Immortals-level rules)
Die, Vecna, Die! (2e Great Wheel Multiverse to world-specific 3e Cosmologies)
"Even with Vecna's removal, his time in the crux effected change in superspace. Though the Lady of Pain attempts to heal the damage, the turmoil spawned by Vecna's time in Sigil cannot be entirely erased. Some Outer Planes drift off and are forever lost, others collide and merge, while at least one Inner Plane runs "aground" on a distant world of the Prime. Moreover, the very nature of the Prime Material Plane itself is altered. Half-worlds like those attached to Tovag Baragu multiply a millionfold, taking on parallel realism in what was before a unified Prime Material Plane. The concept of alternate dimensions rears its metaphorical head, but doesn't yet solidify, and perhaps it never will. New realms, both near and far, are revealed and realms never previously imagined make themselves known. Entities long thought lost emerge once more, while other creatures, both great and small, are inexplicably eradicated. Some common spells begin to work differently. The changes do not occur immediately, but instead are revealed during the subsequent months. However, one thing remains clear: Nothing will ever be the same again."
The Apocalypse Stone (An epic-level adventure for wrapping up a 2e campaign before beginning a 3e campaign)
Chaos War (1e Krynn to SAGA Krynn)
War of Souls (SAGA Krynn to 3e Krynn)
Spellplague (3e Toril to 4e Toril and Abeir)
Dragon #327 "Winning Races: Diaboli--Bringing Diaboli into 3.5E" (describes changes in the Far Realm and Plane of Dreams)
Sundering (4e Toril & Abeir to 5e)
If readers know of any other in-game support for rules-changes, please email me.
(Update: I wrote a post about this here: "The many worlds of D&D? The many multiverses of D&D? The many realities of D&D?" And I found out from "wingsandsword" that the Arcane Age products did give suggestions for how to mimic the pre-Time of Troubles era using the 2E rules, such as different magic. This would be the 2E Reality.)
******
I restate it here: http://www.enworld.org/forum/content.php?4789-Blowing-it-All-Up-and-Starting-Over
What's mind-boggling for me is that from a canonical perspective, each edition of the Rules is a distinct Reality, which has subtle effects on the world itself, which only the gods notice (see the quote the DRAGON magazine article by Bruce Heard):
https://sites.google.com/site/dndphilmont/d-d-realities
Though hardly anyone knows about that, it basically means that the 1E, 2E, 3E, 4E, 5E FR products were each depicting a separate Reality.
WotC bumped into this conundrum when it released the Arcane Age products for 2E which were set in the ancient past...which means those adventures should've been played with 1E rules! AFAIR there was even a paragraph explaining this. But of course the product used 2E rules, which means the "2E Reality" always existed side-by-side with the "1E Reality". Conceivably, each of the five Realities continues to exist, and always existed.
Because the "Rules Realities" aren't "parallel storylines", they presumably model the in-story events from all eras as closely as possible, but using their own "rules lense". So, though no WotC product would actually depict it (because it would confuse customers), the ToT, DVD!, SP, and 2ndS events took place simultaneously in all Five Realities.
For example, in the continuing 1E Reality, the same story-events which lead to the death of all Assassins still happened, it's just that the other classes continued on in their 1e version. When Assassins later emerged again in the FR, whether as a 2E Kit or 3E prestige class, this would be modelled with 1E rules.
Over in the 3E Reality, Assassins were always just depicted as a Prestige Class. Since there are subtle in-story differences between the powers of a 1E assassin and a 3E assassin, there are subtle differences between the two Realities...but only the gods notice.
The Second Sundering is taking place in the 1E, 2E, 3E, and 4E Realities of the Realms too.
I propose Mike Mearls and team bring this Reality concept back to light. It really provides a graspable conceptual framework.
The "6E Realms" and "7E Realms" invisibly exists already too, throughout the past, present, and future.