Not so. Tuberculosis, for example, is tested for that way, and guess what? It is evidence that your other comments are absolutely correct.
Yep, you're right, giving ANY vaccine builds antibodies (that's its purpose, after all) and yep, if you're testing for antibodies you'll pop positive even if you've never been exposed to the actual etiological agent. There is another test for HIV, virus titer, which is an actual count of viral bodies present in body fluids, and it it MUCH more time-consuming and expensive than the antibody test. Far too much so to imagine it replacing that test in third-world countries, for example.
HIV is a toughie for a vaccine because the virus attacks the body's very mechanism for producing the antibodies that the vaccine is trying to increase. I hope they do succeed, but it's a formidable problem. Nor would I necessarily take David Baltimore's word for it - he, after all, is the guy who tried to claim credit for isolating HIV (he called it HTLV), instead of the French fellow who actually did isolate it at the Pasteur Institute. Baltimore's much more at home at the Durban racism congress than in a laboratory, IMHO.
You're thinking of Robert Gallo, the so-called "Pope of AIDS." Baltimore has had his own brush with controversy and allegations of scientific misconduct, but they don't pertain to AIDS.