I doubt the “Ordinary/Extraordinary” terminology will ever catch on. It works well on a Papal document, but the wording is probably too technical and bulky for most people to get comfortable with.
If we want to be historically correct then we could call it the “Gregorian Mass”, or the “Liturgy of St. Gregory the Great” if we want to link it to the Eastern Liturgies. The misleading term “Tridentine” will probably remain the standard, though at least its the most familiar.
The “Traditional Latin Mass” or simply the “Traditional Mass” will also remain common, though personally I prefer the “Classical Latin Mass”. Of course the simplified form of this, the “Latin Mass”, will also remain common, but it is another misleading term since the Novus Ordo can be celebrated in Latin (the Novus Ordo is ordinarily celebrated in Latin- vernacular is by indult).
The “Old/New” classification will probably still be used, as will the “Pre-concillar/Post-concillar” even though that is not entirely correct.
What I would prefer though is a system of terminology that all the Liturgical Rites can fit into. I propose a system that does this, while being faithful to the classification the Holy See uses. Recall the “Anglican Use of the Roman Rite” that was created for High Church Anglicans who converted to Catholicism:
Gregorian Use of the Roman Rite
Pauline Use of the Roman Rite
Anglican Use of the Roman Rite
Ambrosian Use of the Roman Rite
Mozarabic Use of the Roman Rite