Bay City is the real name of the city. Bang City is the derogative name that the inhabitants have given it due to the high crime rate. It's explained in the game.
Ah, OK. I guess I'd forgotten that. I saw those things out of order. I
first ran into Kate saying that, and
then I got this clip from the MC reading the journal.
So there might still be some logic that should be applied here. Something like, the MC can't get that scene from Kate until he first reads the journal. Or, the first time either of those references appears (where the proper name of Bay City is used), the MC has the above thought. And then, if he's already thought that, the thought text is skipped instead of being shown a second time.