(covers information from several alternate timelines)
A warp-capable vessel or warp ship was a term used to describe any type of spacecraft that was capable of warp drive. This capability generally made the distinction between spaceships used space-faring civilizations and starships of warp-capable civilizations.
In his 2063 commencement address, Zefram Cochrane mentioned the first flight of his warp ship, the Phoenix which opened doors for all of Humanity. (ENT: "Regeneration") The term was also used by several Starfleet officers in the 22nd and 24th century describing Cochrane's "experimental warp ship". (ENT: "Future Tense"; Star Trek: First Contact)
In 2152, Esaak, an individual from the pre-warp Valakian species who was attempting to find the cure for a disease that was wiping out their species, asked Jonathan Archer for access to Earth's advanced technology, stating that "With a warp-capable starship, our chances would be greatly improved [for finding a cure]." (ENT: "Dear Doctor")
The term was also used when Captain Jonathan Archer asked Kuroda Lor-ehn if he ever piloted a warp ship and when Commander Trip Tucker asked why Enterprise NX-01 had to transport the Tellarite ambassadors, as this species had warp ships themselves. (ENT: "Canamar", "Babel One")