Two different questions in one there from my perspective.
An "Amiga" (capital A) is computer containing a 68k cpu and one of ocs/ecs/aga.
That one is pretty clear cut, but "amiga" (lower case a) can be any one of a plethora of things, although it basically comes down to the way the system works. OS4, MOS, AROS, and Amithlon are all equally "amiga" to me, as is the minimig. UAE to a lesser degree, but it depends on how a person uses it. Some UAE users I consider to be amigans, others not so much.
Anything post commodore falls under the monikor of "amiga". Very much a parallel to *nix these days. It's simply a word to suggest a certain paradiagm of computing.