"SF isn't north enough to be Northern California" is a common view of people who live in Eureka, Redding, etc., but in my experience people who live in SF think of it as northern California. California basically has two big centers of population: SoCal is LA and friends, and NorCal is SF and friends. Above that are remote regions heard of in legends (either old legends about the gold rush, or newer legends about pot farming).

The Bay Area is definitely not within Central California, taken as a sociogeographic term. The Central California Coast is the coastal region between roughly Santa Barbara and Point Lobos, centered on SLO. There is also another inland definition of Central California: https://en.wikipedia.org/wiki/Central_California