...And the only time any of the Enterprises is considered a flagship, it is in one of these alternate senses.
The E-D is "Flagship of the Federation" in a couple of episodes and the movie ST:Generations. That's obviously not a military concept at all, because of the "Federation" rather than "Starfleet" bit there. Rather, the E-D apparently is the showpiece vessel of the Federation, being sent to impress the natives in various diplomatic contexts and whenever a bully needs bullying back; we see this happen often in TNG.
In contrast, Kirk's Enterprises never were called the flagship of anything.
"Our newest flagship deserves more pomp and circumstace than we can afford today." - Captain Pike, Star Trek