[rossjudson]

Perhaps you should re-read. On thumb-over and thumb-under (TO, TU):

"Both methods are required to play the scale but each is needed under different circumstances; TO is needed for fast, difficult passages and TU is useful for slow, legato passages, or when notes need to be held while playing other notes."

"Beginners should be taught TU first because it is needed for slow passages and takes longer to learn. The TO method should be taught as soon as faster scales are needed, within the first two years of lessons."

[MrGando]

It's both the way the technique is described and the name given to it "thumb over" that are unclear and misleading. I know the technique, and in my years of practice (semi-pro) I've never seen it described anything like that. It doesn't even have a "name" per-se, because you learn to play the piano with your whole body, so it becomes a natural thing that you just have to do to get that speed. And by the way, you can still use it when playing slow, if you want to obtain a certain "tone" or "sound".

[packjaddison]

In my head, after 'thumb-under', I think of 'skips' and 'leaps' depending on distance for whole-hand repositioning, which is descriptively accurate for fingering on the piano, guitar, and violin (etc.). Though, for the novice I could see how conceptualizing a movement this way could interfere with legato technique, as it encourages more discrete chunking than does the 'unbroken' thumb-under during a run. Does that nomenclature match your conceptualization at all?