I grew up, live in, and currently study in the Bay Area. If I am talking to someone, or if someone is talking to me, most references to "the city" will be clearly understood to mean San Francisco. Ex: I'm going up to the city. How was the city? I don't make it into the city that often. There's a concert in the city.
And the fact remains that the majority of the tech ecosystem is centered around the Bay Area, and hence the majority of TechCrunch's target audience.
I find it's best just to call it "San Fran" to disambiguate.
And for the entertainment value of indignant inhabitants of the area gasping like goldfish out of water as they prepare to tell you that no no no, it's not 'San Fran' and only someone completely ignorant of the most revered status of their fair (or foggy) city would make such a horrible faux pas :-)
Not a problem, no one cares what you call SF any more. We'll just know that you're not from around here. And that's ok; we like meeting new people with strange and interesting ways of saying things.
And the fact remains that the majority of the tech ecosystem is centered around the Bay Area, and hence the majority of TechCrunch's target audience.