I consider a starship to be any commissioned Federation Starfleet vessel bearing a warp (or other faster-than-light) drive, a "U.S.S." title prefix, and its own unique "NCC" or "NX" registry.
Any ship no longer commissioned into Federation Starfleet service -- say, an Oberth-class ship sold to the University of Alpha Centauri for long-term, long-range research projects -- would no longer be considered a starship. Support vessels without their own commissions -- say, the Delta Flyer or the Type-6 Shuttlecraft Goddard -- would not be considered starships. Civilian vessels would not be starships.
I believe Picard referred to a Romulan Warbird as a starship in "Tin Man".
Also, based on your definition, would the Vico
not be considered a starshp then? (Was it ever referred to as such in "Hero Worship"? I don't remember.)