As far as I know, the flagship thing only applied to the Enterprise-D
of TNG. Certainly, I think that's the only one we ever heard mentioned that way. Off the top of my head, I know Picard referred to it that way in Generations
when, speaking of Lursa and B'Etor, he said: "They're just trying to decide whether a twenty year-old Klingon Bird of Prey can be a match for the Federation flagship."
Obviously, the term "flagship" is not being used in the traditional sense, since it's not commanded by a flag officer and isn't leading a fleet. Rather, it seems to be used more in the context of being the most prominent and prestigious and, for a time, most powerful ship in the Starfleet.