in-universe, the term was used generically in the 22nd century according to Enterprise and in the mid-23rd according to the 2009 film
We can't tell the difference between generic and specific, because we only follow the adventures of one type of starship in ENT, and only see minor differences between the two vessels called "starship" in STXI (namely, the Kelvin
and the Enterprise
). There's nothing to establish the lack of non-starships in these eras.
Nor is there any pressing reason to abandon the idea of starship as a specific designator. Star Trek is famous for its use of fantastic terminology that has got nothing to do with the real world, and that is exactly as it should be. They have transporters which are not generic means of transport but a very specific technology. Who are we to say that their starships absolutely must be a generic means of star travel?
All that is sort of moot anyway because TOS makes it amply clear that starships stand apart from other types of vessel in Starfleet service. It's even a massive plot point in "Court Martial".
It's like having a truck whose model name is Truck.
Naval battles have been fought with ships called battleships. That's in no way different from using "starship" in a specific way, despite the fact that all ships sailing between stars could be called that theoretically, and all ships engaging in fighting could be justly called battleships.