^I think the idea in TOS was that "starship" was analogous to "capital ship" or "ship of the line" -- a large, powerful vessel rather than a small private craft.
And no, we can't take the "Starship Class" plaque too literally because they were making it up as they went. There's a lot of sloppy and contradictory terminology in the early episodes -- it took them a long time to decide whether the crew worked for Earth, UESPA, Space Central, or whatever, the Federation wasn't even established until roughly halfway through the first season, Vulcans were Vulcanians at first, dilithium was lithium, etc. It should be obvious that it's foolish to take every detail literally at that early stage. These are rough drafts of a universe in progress. If you're writing a novel, you can go back and rewrite your early ideas in favor of the better ones you had later; but in an ongoing series like a TV show or a comic book, you're stuck with your early ideas, so the best you can do is retcon them and ignore them and hope your audience will suspend disbelief and accept the revised version as the "true" one.