The Tridentine Mass and the Novus Ordo Mass are both forms of the Roman/Latin Rite. They are not separate Rites.
Yes, you are right! For purposes of clarification, however, the better word would be 'liturgy'. In the 'liturgy' of the Mass of the 1962 Missal, the Prayer to St. Michael the Archangel is still said. In the 'liturgy of the Mass of the 1969 Missal, the prayer is no longer included.