I'd say Starship class was used in the 2240s for the 17th cruiser design when the class was launched as Starfleet's state of the art ship of the line. These vessels were so prestigous that their commanding officers weren't simply captains but "Starship Fleet Captains" (Garth of Izar was referred to as such and it probably got later shortened to "Fleet Captain").
But then the 18th cruiser design and the 19th cruiser design were launched and the 17-cruisers weren't that special any more and everyone started to call them by their other class name.
But then the 18th cruiser design and the 19th cruiser design were launched and the 17-cruisers weren't that special any more and everyone started to call them by their other class name.