I'd say an Amiga is a computer with a hardware design developed from the original Amiga and running (or at least capable of running) an OS developed from the original Amiga OS, and which is capable of running most software (allowing for differences in OCS/ECS/AGA) written for the classic Amigas without having to run an Amiga emulator. So, as far as I'm concerned that means only the machines made by Commodore and Escom, and of course the pre-Commodore development systems (Joe Pillow) built by Amiga.
The OS part of it is a bit fuzzy... a real Amiga needs to have the hardware part of the OS (KickStart ROMs), but while they generally run the Amiga OS (Workbench) they can run other OSes like Amiga Unix, Linux, or even the various custom "OSes" that some NDOS: game/demo floppies use.
If some company were to go back and try to develop a new system by updating/extending the old hardware design which was backwards compatible with Amiga software including the OS, that would kick ass. I doubt it will ever happen though.
The companies calling themselves Amiga post-Escom are just people buying and exploiting a name, selling products with almost no connection to the original. Say you have a good friend named Joe. You really like Joe, he's a reliable and interesting guy. Joe gets drunk and jumps off a 20 storey building to his death. Bob is walking down the street and sees the remains of Joe's body bleeding on the sidewalk. Bob whips out a hunting knife and skins Joe, removing his skin and leaving the bloody rotting carcass on the sidewalk. Bob then makes a suit out of Joe's skin and shows up at your house saying "Hi, I'm Joe. Let me come in and hang out with you." That's pretty much what Amiga Inc. is today.