Regarding smaller ships not being considered "ships", I think DS9 reliably resolved that conundrum via the runabouts. And there are just as small ships in the real world that are still designated as fully commissioned naval vessels (though this is rare in larger navies). However, given the established role (or rather, multirole capabilities) of the Danube class, they may very well be exceptions for ships that small.
Additionally, the Nash (DS9) establishes that something as simple as a transport can still be a fully commissioned starship.
The definition of a starship changed somewhat between TOS and TNG, where it was earlier referred to as a special type of vessel (separate from cruisers and freighters), to it ultimately becoming a more generic term for any long-range vessel.
Even so, I think there are vehicles in Starfleet's inventory that do not classify as "starships," but rather as "craft" due to being small and/or having a very limited operational range.