Here in the Bay Area, calling San Francisco The City is a relic from the days when it really was the only city worthy of the name within several hundred miles. I grew up in the Santa Clara Valley before it was renamed, and ‘going to the City’ meant dressing up in your best clothes so as to be presentable at the play, concert, museum, fancy hotel, or fancy restaurant – not one of which was available outside San Francisco.
Calling it Frisco is a mark of a tourist who is probably from Nebraska.
Everybody in every city everywhere refers to it as The City.
Cutesy nicknames for cities are never something used by the locals. I have never once heard anybody from Pittsburgh actually call it “Da Burgh.” If I did, I’d assume he was trying to be funny.
I won’t try to answer the OP. But I will say that in my experience with Bay Area folks, and particularly those living in San Francisco, using the term “Frisco” is inappropriate.
For most of my life I was a SoCal guy. My uncle Don, a born-and-bred San Franciscan, told me shortly before I graduated college to avoid using “Frisco.” He felt it demeaning. But his intention was to make me realize that the fastest way to lose the respect of my SF audience was to appear as a rube. I took it to heart. For me, San Francisco is “The City,” “SF,” or the full name.
It’s 40 years later and I still adhere to his advice.
(BTW: I like it that notwithstanding Jack Webb, folks in Los Angeles don’t refer to L.A. as “the city.” We [now former] Angelenos call it L.A. and are proud of it.)