The "Bay Area" is well known to refer (only) to the San Francisco Bay, while the "Greater Bay Area" is known to only refer to the Pearl River Delta Bay (HK, Mac

The "Bay Area" is well known to refer (only) to the San Francisco Bay, while the "Greater Bay Area" is known to only refer to the Pearl River Delta Bay (HK, Macau, Guangdong). Other bays are typically named more specifically (e.g. Tokyo Bay, Monterey Bay, etc.). You'd be more on point if OP had referred to San Francisco as "The City" as many in the City do; since that's something that's also used in other cities like NYC. NB: current resident of the Greater Bay Area in China and former resident of the lesser Bay Area in California.