Geordi's use of the phrase "The flagship of the Federation" was correct when referring to the NCC-1701-E in Star Trek: First Contact because the Sovereign-class Enterprise was the most advanced at the time.
Flagship can also be used to refer to a ship when it is the lead ship in a fleet of vessels, a designation given on account of it being the largest, fastest, newest, most advanced, most heavily armed or, for publicity purposes, the most well known.
Okay, you're quoting the bolded portion directly from Wikipedia's "Flagship" entry (or rather, you're plagiarizing it -- in future, don't quote something directly without citing your source, please). But that's not always the most reliable source, and I think it's conflating multiple different uses. In strictly military terms, I think Babaganoosh
is right -- a flagship is the ship on which the commander of a fleet (whether a group of ships operating jointly or simply the navy at large) is based. Its use to mean the grandest or most advanced ship in a line is more a civilian usage, as in a cruise fleet, say.
Of course, Starfleet has never been strictly military, and usage can evolve over time. Still, the "flagship" designation has never been clearly defined in Trek, and I suspect it's an informal usage.
Regardless of how you define it, I don't think the Aventine
would be the "flagship." It's more of an experimental ship, a test bed for new technologies. The "flagship" of the fleet might get the best, most advanced new toys, but not until after the Aventine
has taken them for a test drive and made sure they won't blow up.