Probably because LA County has over 75% of the population of its MSA and over 50% of the population of its CSA; by contrast, however you define it (its MSA, the traditional 9 county "Bay Area", or its CSA), San Francisco County is only a minority of the population of its metro area. It isn't even the most populous county in the Bay Area–Santa Clara, Alameda and Contra Costa all have significantly larger populations. Similarly Harris County and the City of Houston dominate the population of their metro area in a way in which the City and County of San Francisco does not.

I still think there is a difference here between American and Australian usage – American popular conceptions of geography pay far greater attention to local government boundaries than Australians do. That makes complete sense, considering that local governments in the US have vastly more power than those in Australia – here, courts, elections (even local elections), prosecutors, corrections departments, law enforcement, fire departments, ambulances/paramedics, public schools, public hospitals, public transport and land titles are all exclusively state government responsibilities, and in inner metropolitan areas, the state government runs the water supply and sewerage system as well. Americans pay much more mental attention to local government boundaries than Australians do, because those boundaries have a significantly greater impact on their everyday lives.

I've always considered it to depend mostly on whether we're talking about the current location or not.

So if we're actually in San Francisco metro area and I talk about "San Francisco", I probably mean the city proper. But if we're in DC and I talk about "San Francisco", I probably mean the metro area. Or, commonly, I actually mean somewhere specific but I don't expect my listener to know the more specific areas, so I just say "San Francisco".

I tell people from out of state that I’m from Chicago, because they’ve heard of that but not Geneva, Illinois.