Likewise, the Captain's Yacht should have been the Captain's Gig.
Since the thing never saw use in TNG, we might just as well decide it was indeed the Captain's very own civilian yacht, docked to his starship because rank hath its privileges.
Heck, the E-E yacht had a dedication plaque identifying Picard by name - an unlikely feature in a craft supposed to be part of the permanent complement of a starship where commanding officers come and go, but quite to be expected aboard a private craft belonging to a single person!
Timo Saloniemi