I'm not so sure. Most people I know have the idea that if it acts like UNIX they can call it a UNIX. Your definition also makes OS X not UNIX, which I think we both agree is UNIX. I think who owns the rights makes the rules, so only four operating systems get to be called UNIX. The rest are "UNIX-like" or "POSIX-compliant."
Linux, like the Minix clone which also came from Europe, are completely separate families with different originators, and different derivatives.
At least now you appear to be disagreeing with the charlatans who say Linux has ripped-off AT&T UNIX code.
The NeXT systems were by far the nicest workstation or desktop computers I had ever seen.
I know some people in academia who still use them. Awesome machines, the WWW was invented on one because current proprietary text repository systems had onorous licensing.
Huh? You're the one confusing things.
From the first line on Wikipedia for "BSD":
Historically, BSD has been considered as a branch of UNIX — "BSD UNIX", because it shared the initial codebase and design with the original AT&T UNIX operating system.