Indeed how? If they really mean SF, then write that. Bay Area is much greater and much more than SF. San Jose alone is larger in area and people than SF and Bay Area is much much more than San Jose.

I'm trying to spread awareness to people not from here that the two things aren't the same at all. It's bad enough that media convulsively regurgitate the same tired fallacy repeatedly.

Perhaps if you consider that when people speak of the Bay Area, this is sometimes an example of metonymy (or even in another sense, synedoche), you wouldn't feel the need raise everyone's awareness of San Jose.

“Cultural hub” is a term for “shows up in movies”. It’s completely irrelevant if you live in the South Bay.

Well, it's not like the South Bay has culture.

Relevant video response/blast from the past (there is also a 2015 sequel): https://www.youtube.com/watch?v=SlTvSUCCqPo

…that you like.