Now, interestingly, I'm not sure we ever heard the term "flagship" applied to any version of the Enterprise other than the D. It certainly was never used in TOS or even the TOS movies that I recall.
Actually, Pike did call the Enterprise
"our newest flagship" in the 2009 movie. Then again, this was the same guy who called the Federation an armada...
Also, the Mirror Universe NX-01 was called "the Terran flagship" in "In a Mirror, Darkly." Archer subsequently called the captured Defiant
"my flagship," but he was thinking of himself as an emperor-to-be at that point. That implies that both conflicting uses of "flagship" were in use there. Although I think the mirror NX-01 was seen as part of larger battle groups here and there, and it was engaged in an active war, so maybe it was the "Terran flagship" in the sense that it was the command ship of the armada in joint operations.
Which makes me wonder if Starfleet considered the B, or the C, or the E to be their flagship?
has comm chatter referring to a ship other than the E-E as the flagship, but that's during the battle with the Borg cube so it's probably being used in the sense of the command ship of the battle group.