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.
That's not correct. The original classification according to The Making of Star Trek
was "Cruiser Class" which evolved
into "Starship Class" which remained and stayed.
Instead "cruiser" apparently became a type according to this Jefferies sketch
(funny, looks like "space cruiser" suggests there are still naval cruisers around, hence the differentiation).