I just figured out the different right when I posted the question. lolz.
Amphoteric is not necessary to be amphiprotic as amphiprotic involves proton accepting and donating while amphoteric relate to Lewis acids and bases (electron pair accepting and donating respectively). Thus, I've got a sort of diagram
A --> amphiprotic --(Bronsted-Lowry)--> amphoteric
A --> amphoteric --(Bronsted Lowry)--> amphiprotic
or ......................--(Lewis)--> st else.
This is quite cool, isn't it? I think this is similar to your explanation. Thanks anyway.