I'm not even sure they had starships 200 years before the events of WNMHGB. Enterprise fans, help me out here.
This is quite tangential to the thread at hand, but yes, in the broader Trek universe context, ENT "retroactively" confirms that the term "starship" was already being used for both foreign and domestic spacegoing vessels in the 2150s; the hero ship of the show was of that designation, too.
TOS itself refers to past spacecraft such as the
Archon as "starships", which is only logical, as nothing indicates that the class of ships to which Kirk's first
Enterprise belonged would have been the very first starship class ever. It follows that by the time of TOS, there probably were several classes of starship in operation: "current", "past" and "future" (and not necessarily just a single class in each category). This is also kinda supported by "Court Martial" where the famous Star Ship Status chart lists registries in the 1600, 1700 and 1800 ranges.
In "Doomsday Machine", Spock uses the phrase "by configuration, a starship stopped in space". This can be taken to mean that all starships share a "configuration", whatever that is. Or it can be taken to mean that the "configuration" of the
Constellation fit within the diverse library of known starship "configurations".
All in all, it seems a bit unsatisfactory to try and argue that Kirk's (baker's?) dozen would be the entire Starfleet arsenal of starships as of the late 2260s. Whenever starships are name-dropped, we should be free to decide whether they are part of the bunch Kirk mentioned or not, unless compelling visual evidence points us in one direction or another.
Timo Saloniemi