Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
Albatrosses start out with mutual grooming, each tenderly preening the other bird's feathers. Then they launch into a mating dance where they alternate between tapping their beaks, opening their mouths, and looking at the ground. To a casual observer, it looks like the birds are jousting, their beaks rattling together like castanets. 8. Blue Manakin