As a result of the first principle (shulchan melachim), French fries and potato chips do not require bishul Yisroel, since those are foods which are not served at more sophisticated events. [That said, these items require hashgachah to ensure that the oil, other ingredients, and frying equipment is kosher.] In contrast, IQF (individually quick frozen) potato products which are fully cooked, such as frozen baked potato wedges, require bishul Yisroel at the factory since they are deemed suitable for shulchan melachim.
In many cases, the French fries or potato chips are seasoned after they are fried. In that case, the “base” product might be certified as kosher pareve, but the seasoned or flavored varieties might be dairy or even non-kosher. In some cases, it is not feasible to kasher the equipment between different seasonings – due to the heat of the product, the difficulty in cleaning the tumblers used for seasoning, or the expense in bringing a Mashgiach to oversee the process – and therefore all seasoned chips or fries will be labeled as non-kosher or as dairy.
The second principle (cooking with steam) applies to instant potato products such as potato flakes and granules. The cooking of those products occurs in a large tank where the potatoes are blasted with direct, live steam until they are fully edible. [The resulting “mashed potatoes” are then dried out to create instant potatoes.] Since the cooking is performed with steam, bishul Yisroel is not required. [Before cooking, the potatoes are blanched for a few minutes in hot water, but in this case, the blanching typically does not render the potatoes as passably edible.] That said, these items require hashgachah to ensure that the minor ingredients added, and the equipment used for drying etc., are kosher.
The third principle (passably edible) affects two classes of potato products. The first is dehydrated potatoes, such as potato slices and dices, potato shreds, and hash browns. The blanching done to these items typically is enough to render them passably edible, which means that if a Jew does not participate in that step of the process, the dehydrated potatoes will be forbidden as bishul akum. Nonetheless, some reputable hashgachos allow these items to be sold as “certified” even though they are actually bishul akum, because they are expecting the end user (restaurant or consumer) to have a Jew finish off the cooking and reverse the bishul akum status. [The cRc would not certify dehydrated potatoes under such circumstances.] This means, that the certified dehydrated potatoes are actually non-kosher when they come out of the package, and – until a Jew finishes off the cooking – one should not put them into a salad, add to an already-cooked soup, or nibble on them.
An additional case where the third principle is a consideration is that of canned potatoes. These items are fully cooked, but it is common that the end user will “brown” them before serving them. Since the item is fully cooked (as opposed to being just passably edible), it is “too late” for the Jew to reverse any existing bishul akum concern. Accordingly, it is generally accepted that canned potatoes are only kosher if a Jew participated in the cooking/canning stage.