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."
Right. Even canon is subject to change, since of course they're making it up as they go. Later intentions should override initial intentions. "Starship Class" is one of those awkward early ideas that were abandoned when something better came along, like lasers instead of phasers, Vulcanians instead of Vulcans, and UESPA instead of Starfleet.
As for why Enterprise
couldn't also be a "Starship Class" vessel, the answer is that that term is simply too generic. Maybe at the start, TOS's creators were treating "starship" as the equivalent of "capital ship," a term only used for the larger vessels, but the term has been used far too widely since then for far too many types of ship. Basically it's come to mean any interstellar vessel larger than a shuttle or runabout. Calling a Starfleet vessel "Starship Class" would work about as well as calling a US Navy vessel "Ship Class."