But given how much we know about the previous command histories of other captains, I think it's an interesting little oddity of the original series (and movies) that we never got a definitive and unambiguous answer one way or the other onscreen.
Well, '60s shows were more concerned with the here and now for the characters, unless it was convenient for the story to bring in an old friend or lost love or family member we'd never heard of before. We did learn a fair amount about his career as a cadet and junior officer, but nobody ever came up with a story focusing on his first command, just by luck of the draw.
We also know, thanks to ENT, that by the time the NCC-1764 Defiant was built, the plaque now says "Constitution Class" and not "Starship Class." So at some point between the construction of the Enterprise and the loss of the Defiant in "The Tholian Web," the nomenclature was changed (at least for the Defiant).
Or maybe it "really" said "Constitution Class" all along and the set decorators got that detail wrong. Sometimes it's best to treat what we see onscreen as an approximation of the conjectural reality rather than an exactly literal depiction of it. In Gene Roddenberry's preface to his own novelization of ST:TMP, he took on the role of a 23rd-century producer who'd dramatized the "real" experiences of Kirk and his crew, and apologized for having portrayed them in a way that was "inaccurately 'larger than life.'" No doubt there were parts of TOS that he was unhappy with and wanted to renounce, and he was suggesting that the fans should take them as inaccuracies in the depiction of the "real" missions of the Enterprise
. When TMP came out and revealed a new design for the Klingons, Roddenberry asked fans to assume they'd always looked that way and the show just hadn't depicted them accurately. So he wouldn't have wanted his fans to take every detail literally, because frankly a number of the details just didn't work.