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.
Mynah birds have long been a popular bird. In ancient Greece, the mynah was kept among the aristocracy as a pet, and in India, the mynah has been considered sacred for more than 2,000 years, and during a feast day, individual birds were carted through the city by oxen.