My understanding is that UCLA was originally called "University of California, Southern Branch." During a Westwood campus tour, the tour guide gave an accounting of U$C students--or was it C.al students?--disparagingly referring to what-is-now-called-UCLA as the "twig." That story may be apocryphal, but I thought it was pretty funny when I heard it.
I went to grad school at UCLA and now work at UC-Santa Barbara, and, in my experience, the "University of California" and "C.al" are the most common terms for the Berkeley campus. "U.C.-Berkeley" is used, but less frequently. I've also heard the term "Berkeley," which is understood to refer to the east bay city/campus and not the music school in Boston (which is spelled differently).
The two-letter "U.C." is never used to denote the Berkeley campus. "U.C." is used to denote the 10-campus institution known as the UC system. The UC system has a President (Janet Napolitano), and each campus has its own Chancellor.
Well, that's the way the name game works in my world, but I don't really sweat these details. 71Bear or someone else should, of course, feel free to correct any of the above.
Before I go, I'd like to confuse matters even more...
The URL
www.berkeley.edu will take you to the home page of the Berkeley campus,
www.california.edu will take you nowhere, and
www.universityofcalifornia.edu will take you to the home page of the UC System.
Give me
www.stanford.edu any day!