Uniat Church, or Uniats, has a much wider signification than United Greek Church or United Greeks, and embraces all the Eastern Churches in communion with Rome, but following another than the Latin rite, whether it be Byzantine, Armenian, Syrian, Chaldean, Maronite, or Coptic. The Uniat Church is therefore really synonymous with Eastern Churches united to Rome, and Uniats is synonymous with Eastern Christians united with Rome.
Good point katnip.