It's pretty self-evident that any manned vehicle capable of voyaging between star systems is in fact a "starship" even if said vehicle is a relativistic vessel going slower than light. Of course that doesn't mean everyone automatically interprets it that way.
By the early to mid 23rd century the term "starship" could come to mean something more specific in context of TOS. It might have something to do with the ship's intended primary role: extended deep space exploration. By the time of Pike and Kirk the Enterprise
and her sister ships are recognized as something special, something meant to operate more autonomously for extended periods to probe deep into unfamiliar and unexplored territory. They are the Federation's vanguard.
And so the TOS Enterprise
can easily fit into three classifications at the same time.
- Star Ship class (primary mission profile)
- Heavy Cruiser class (type of vessel)
-class (ship's configuration)
When the ship is refitted in TMP we can assume its configuration has changed now to Enterprise
-class because she is the first of her kind even though the other two classifications can still apply.
Sometime after TMP the "Star Ship class" designation falls into disuse for whatever cultural/political/organizational reason. Maybe the Federation pulls back on extensive deep space exploration or some other reason. In TOS' time there might have been a pioneering/exploratory push fuelled by cultural and political reasons as well as great technological advances which fed into the idea of a "starship" being something really special by what it's meant to do and what it represents: the Federations noblest ideals.
Kirk's line of "only twelve like it in the fleet" doesn't mean there are only twelve starships or only twelve star faring ships in existence. It can't credibly mean that. It's more likely his reference means twelve ships specifically designed, fitted and manned to do what no other vessel could do (at that time).
Merick: "He commands not just a spaceship, but a starship. A very special vessel and crew."
Stone: "Not one man in a million can do what you and I have done: command a starship."
What makes Kirk and the Enterprise
and her crew special (as well as the other eleven ships) is because they're meant to be the best of the best assigned to do something seen as grander and more hazardous than anything comparable.
Just my two cents.