Anything other than those twelve vessels weren't even called starships. Back then, when Star Trek was in production, there was no such thing as a "starship" that wasn't one of those twelve ships. The idea that there were multiple classes or "flavors" of starships other than the class of the twelve established starships was an idea cooked up years later--long after Star Trek went off the air.
Pike suffered injuries on a cadet vessel that was an "old Class J Starship
" in "The Menagerie". That would indicate there must be at least one other class of starship other than the one the Enterprise belongs to. Well unless the Enterprise was also an "old class J" which I doubt. And given that there are many letters of the alphabet that could also be used for classification of starships, there could be more variety of classes of starships out there than just two.
Since you invoked Occam's Razor, then in TOS the Enterprise was one of twelve ships in the fleet that belonged to the "Starship Class" (as seen on the plaque). It can't really be argued to use what the script intended simply because the final product didn't air it the way it was intended and they never did swap out that plaque on the bridge with the "intended" class name of "Constitution". And we can't argue for the 1st bird or such either because that also did not ever air in the series, AFAIK. What we're left with is simply "Starship Class". Future history would eventually relabel her "Starship Class" to either a subset of "Constitution Class", or just re-classify her into the "Constitution Class".
My IMHO, of course.