I´m not refering to different expressions of a single kata, but rather the different possible applications that can be done by a (sequence of) waza. I guess that's what you also describe in your question, by: ...maintain the structure and mechanics of the movement shown.
These movements can be slightly alters so a different execution/application appears.
I have the idea that people who claim to 'know' three thousand applications actually mean, that by altering a uraken(backfist)- to a shuto(knifehand)-uchi(strike), they have a different application. Where in my vision, they have merely found a variation. The principle/structure/mechanics of the particular strike remain the same. Only the form of contact alters, which might have influence on the follow-up, but that's a different chapter.
I see kata like this:
A kata is built up in sequences, made as a training aid. All of these sequences are open to variation. (Try for instance: stepping in a different direction; not during the complete kata, but by studying a sequence.) Some techniques are open to variation, because of similarities in structure and/or principle. (Tsuki can be many different hand forms, the same goes for uchi.)
Even consider hikite and soete; what are they in your* karate? *(Not specifically adressed at you Ronin1966, but in general.)
Does altering all shuto-uchi to uraken-uchi in a kata, make it a different kata?