In this case, the early, awkward use of "Starship Class" has long since been superseded by a more plausible usage wherein "starship" is a generic term for any interstellar spacecraft and ships are given class names that are generally taken from the first vessel in the class.
...However, it would not really go against real world precedent to treat the terminology as mutable. Say, HMS Dreadnought
mutated into dreadnought the ship type.
More relevantly to this specific issue, "cutter" mutated from being a type of boat in Coast Guard use (literally, a "boat") into a generic name for all types of boat and ship in Coast Guard use. The exact same thing might have happened to "starship", it starting out as a word similar to "battleship" and indicating a specific type of heavy unit (of which NX-01 was one, and by the standards of that time, so was any vessel she encountered, so we don't even have to worry about retcons!), but then becoming generic either just after TOS, during it, or even some time before it (but old habits die hard, and old brass plates don't get replaced any time soon).
Just saying. The idea that "starship=/=spaceship" or even that "starship>spaceship" is pretty strongly in evidence in TOS and the TOS movies, all the way up to ST4. It warrants at least some sort of support - even if one subscribes (like me) to the sensible Ferengi rule that ultimately, writer intent plus an empty sack is worth the sack.
I agree. The only problem, though, is that the idea that the original Enterprise was "Starship Class" is rather directly contradicted by several later series. The one that springs to mind immediately is "Relics," wherein Picard refers to the original ship's bridge as "Constitution class" and Scotty agrees. Now, you could extend that and say it is a "Constitution class starship," but to say it is a "Constitution class Starship class ship" seems unlikely.
While it is fact that there was a plaque on the Enterprise bridge that did, indeed, say "Starship Class" and not "Constitution class," the plaque is barely visible -- I'm not certain we can ever make out the wording in TOS -- and the lines of dialogue in later series are pretty definitive. Therefore, much like writer intentions that are later contradicted, I feel safe in ignoring the plaque in the same way we ignore "James R. Kirk."