TOS had at least one flagship in the sense of a ship having a flag officer aboard and commanding assets from that position - the Lexington controlled a formation of ships, with Commodore Wesley in charge (although the ship herself may also have had a lowly Captain commanding her own maneuvers, as is common with real-world flagships of this sort).
We don't know if the other starship head honcho of Commodore rank, Matt Decker, also had a formation of ships following his flag. They were suspiciously absent from the episode, but the jamming effect of the Doomsday Machine would have eliminated them from the picture anyway.
In another sense, Commodore Mendez (or at least his illusion) literally broke his flag aboard Kirk's ship, turning her into his flagship. That is, we see two flags flying at the inquiry, and while the light blue one appears to have the Starfleet Banana symbol, the other one in studio reality appears to be the Cuban national flag, and might be Mendez' personal flag in-universe.
TNG made several references to flagships in the sense of ships used for commanding formations of vessels, although we never learned that this would require a special type of vessel. In the real world, generic combat vessels have been used as flagships for a much shorter period of time than dedicated combat-incapable vessels - if one excludes the age of sail. But in the Trek context, one probably shouldn't do that.
As for specific Enterprises...
Archer's ship: very advanced, the first-ever Starfleet deep space explorer. the biggest Starfleet vessel we saw. Never had a flag officer aboard giving orders (excluding the Mirror Universe), never actually commanded a fleet of ships (excluding the "Twilight" alternate timeline), never was referred to as flagship. In practice, was Earth's showcase ship. Probably best defined as "flagship-capable" and potential "United Earth Flagship".
Kirk's first ship before refit: not advanced nor first, but the biggest seen (excluding alternate timelines). Had a flag officer aboard giving orders a couple of times, never commanded a fleet of ships, never was referred to as flagship. Wasn't Starfleet's, Earth's, Federation's or mankind's showcase ship. Sister ships showed prowess at flagshipping, tho. Probably best defined as "flagship-capable" in the strictly military sense.
Kirk's first ship after refit: potentially advanced and first, but not assuredly either of those things, and not the biggest if background graphics of dreadnoughts count. Had a flag officer aboard giving orders once, never commanded a fleet of ships, never was referred to as flagship. Depending on whether the refit gave her novelty value, may have been a Federation showcase ship, and certainly was a notorious Starfleet vessel. Probably best defined as "flagship-capable", but again in the strictly military sense, as Starfleet wouldn't want to make too much of a fuss about her notoriety by actually formally calling her flagship in the "best in show" sense.
Kirk's second ship: Technical specs supposedly same as with the first refitted, but did not have a flag officer aboard at any stage. The other things being equal, probably best defined as above.
Harriman's ship: Biggest ship seen to that date. Never had a flag officer aboard, never commanded a fleet of ships, etc. - but we saw so little of her that all bets really are off. Probably best defined as eminently "flagship-capable" both in the military and symbolic senses.
Garrett's ship: Biggest ship seen to that date. Never had a flag officer aboard etc. - but again we saw very little of her, so as above.
Picard's first ship: Again biggest ship seen to that date. Had flag officers aboard, and Pressman attempted to assume command, while Jameson assumed mission command. Was supposed to lead fleets of ships under Jellico at least. Was called the Federation Flagship twice. Probably best defined as "flagship-capable" in the military sense, and definitely as "Federation Flagship" in the showpiece sense.
Picard's second ship: Not as big as the previous E, but was referred to as the most advanced. The one flag officer seen aboard did not try to assume command, but a fleet of ships was briefly commanded by Picard himself. Never was called the flagship of anything. Probably at most defined as "flagship-capable", then - although by that date, even midgets like Defiant could serve in that role, so it's not saying much.
As for the name Enterprise only going to important ships, this doesn't appear to be the case. Between Archer's ship and Kirk's, Earth and Starfleet both felt that the best ship should be named something else altogether; likewise between Garrett's and Picard's ships, and potentially between Harriman's and Garrett's as well - or even Kirk's first and Harriman's, since the name did linger but on an aged design. On the other hand, the upward trend in size was lost at the transition from D to E, so the name might be fading to obscurity again.
Timo Saloniemi
