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".
I would take that as Picard either speaking imprecisely, or as Picard applying the same label to a ship commissioned into the service of the Romulan Imperial Fleet in the same way it's used for their Starfleet counterparts.
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.)
I don't remember if it was, but I seem to recall that it was called the S.S. Vico
and had an "NAR" registry. So I'm inclined to interpret the S.S. Vico
as not formally being a starship.
Again, that's just my interpretation of things.