Yes, in general people do know "Bay Area" means the SF area.
I'll blow your mind one further and say that a lot of people on the west side of the country will recognize the nickname "The City" as referring to San Francisco, as well, even if they live in other cities on the West Coast. For a very long time, SF was the only major city west of the Mississippi that could match the East Coast cities in size, culture, and technology, and so if you said "The City" out west, you automatically and obviously meant SF. It wasn't until Los Angeles started to really grow in the early 20th century that SF had any kind of competition in that regard. To this day, it retains that nickname among older folks, and I even knew people who lived in the Seattle area who meant SF when they said "I'm going to visit The City for a few days."