But the converse is not so: if you intend to go to another star system, it doesn't necessarily follow that it will be FTL. So it's a mistake to use FTL as part of the definition. The term "starship," in general usage, refers quite simply to any sizeable, crewed spacecraft intended for interstellar travel. That's true whether it uses some hypothetical form of effective superluminal travel such as a warp drive or more conventional, slower-than-light means such as a ramjet or lightsail system. There's lots of science fiction and scientific speculation out there about slower-than-light starships. I tend to take the distinction in Trek between "starship" and "spaceship" as analogous to the distinction between "ship" and "boat." A seagoing vessel is a ship if it's large enough and independent enough, or if it carries another vessel. Although I definitely think it's a sloppy use of the term to refer to a runabout as a starship.
I've taken the position that the term "starship" changed between TOS and TNG, and went from being special to generic. I think the only reason why a warp-capable shuttle might not be considered a starship is due to its small size (if a runabout was perhaps several meters shorter in length, it might have been called a long-range shuttle instead, IMO).