If you're curious about what the Jump Skill does and not the mechanics behind jumping in general, please see the Horse Skills page.
Obstacle Height: The height at which your horse jumps is primarily based upon the obstacle size in front of it (Low, Medium, Tall). Obstacle height will also change the landing distance. Tall obstacles require more room for the landing than Low obstacles since the horse has to descend from a higher altitude. More on this topic in the below section Jump Height and Stamina Regeneration.
Horse Breed: Animation speed does not change between breeds, but, The Irish Draught's jump is slightly higher and travels further than the Dutch Warmblood's because their model is a bit larger. While I haven't noticed the jump height difference being meaningful, the horizontal distance absolutely is (see Figure 1-A., Figure 1-B. and Figure 1-C. below). Keep this in mind when designing and running courses, and remember to take advantage of each breed's strengths:
Irish Draught Scale: 1.015
Dutch Warmblood Scale: 0.97
Speed: Horse speed, as shown in the interface, will affect the horizontal distance covered by the jump. More on speed changes in the How Do Mid-Air Speed Adjustments Work? section.
Figure 1-A. Difference in landing position between the Irish Draught (left) and Dutch Warmblood (right) at the Working Canter over a Tall jump.
Figure 1-B. Comparison of the the Irish Draught's (left) and Dutch Warmblood's (right) jump at the Working Canter over a Tall jump.
Figure 1-C. Practical example of the differences between the Irish Draught and Dutch Warmblood (both with 50 in temperament) in the Advanced Competition. Both horses clear the distance from obstacle 10 to obstacle 11 in three strides, but the Irish Draught has a significantly easier time since his ground coverage makes the distance more forgiving in this scenario. While the Dutch Warmblood was capable, it would have been much easier to add in an additional stride. Note that this footage was recorded in the 0.1.0.5 version of the game, but the example is still applicable.
As previously mentioned, the height of the obstacle you're jumping affects how long you're in the air and your landing position (see Figure 2-A.). The higher the obstacle, the further away from the obstacle you will land. However, stamina regeneration always begins around 0.86 seconds (860 ms) after hitting the Spacebar, regardless of jump height (Figure 2-B. below).
Figure 2-A. Difference in landing position and air-time between the Low (0.85 sec, 850 ms), Medium (0.92 sec, 920 ms) and Tall (0.99 sec, 990 ms) jump heights.
Figure 2-B. Difference in landing position and air-time between the Low, Medium and Tall jump heights. Stamina regeneration timer shown as well.
If you've played Tales of Rein Ravine, you've probably found yourself wondering if there is an ideal time within the jump window to gain an extra bonus to your jump power in Fernhoof Grove. The answer is NO. Jessi has confirmed that all parts of the jump window are created equal with regards to jump power.
However, I wanted to do some additional related testing...
I compared the animations of jumping at the very beginning of the stride window to the animations of jumping at the very end of the stride window, and the animation of both jumps synced up within 0.14 seconds (140 ms, 16 frames at 120fps) and stamina regeneration began at exactly the same time. There doesn't seem to be any benefit to jumping at a particular time within the jump window since the animation syncs up so quickly, but, of course, there's still more science to be done. For now though, I think it's safe to assume that it doesn't matter when you jump within the window.. just be in the window (see Figure 3.).
In Summary: As far as I can tell, as long as you hit Spacebar within the jump window, you're getting the most power and advantage possible.
Figure 3. Comparison of jumping early in the jump window vs late in the jump window.
The horizontal distance covered by the jump is decided by the speed shown on the interface at the time of jumping (when you hit Spacebar). With this recent update (v0.1.0.6), your speed can also be adjusted slightly while in the air by speeding up with W or slowing down with S (see Figure 4.). Speeding and slowing mid-air does NOT WORK if you use the Shift key to do a quick transition. If you use the Shift key, your horse will transition upon landing like in previous versions.
So, if you want to adjust your speed mid-air, make sure you're using only W and S to speed or slow. Use Shift if you're wanting your speed to adjust after you've landed. Both of these methods are useful in different situations, so pick the one that suits your needs.
Figure 4. Mid-air speed adjustment comparison.