In Abrams's first Trek movie, Pike describes the Federation as "a humanitarian and peacekeeping armada."
That covers it, period.
They may be military, but they don't think of themselves primarily as warriors (Kirk: "We've been trained to think in terms other than war," in the episode "Day Of The Dove") or their vessels as warships.
"The Squire Of Gothos:"
KIRK: Our missions are peaceful, not for conquest. When we do battle, it is only because we have no choice.
You can Google any definitions you like, but you'd be hard pressed to find the Enterprise or other Starfleet vessel described by its crew as a "warship" in TOS.
Sounds like the writers did their research and were less lazy than many fans who are sure they know TOS backward and forward.
I'm certainly glad to know that the writers of these movies are actually taking pains to make the distinction that's always been made in Star Trek between the exploratory and sometimes necessarily defensive missions of Starfleet and a fleet of warships - we do want all these millions of new fans to pick up on that difference, after all. "Gene's vision" and all that. You start treating the Enterprise as a warship and you're diminishing the essential differences between Star Trek
and Star Wars.