I'm confused. The article that you linked says San Francisco is the most expensive rental market. However, you speak about UC Santa Cruz. The campus is about 120 km south. It's a totally different rental market. Do I misunderstand?
It’s an hour and 14 minute drive. It’s not unheard of to commute from there, and it’s far more beautiful than any of the closer beaches, so yes the Bay Area rental prices still apply, in addition to beach community rental prices. As far as calling it SF, some people shorten “The San Francisco Bay Area,” which includes San Jose, to “San Francisco,” and people still know what they mean, though I see why it can be confusing.