But the Vatican doe not officially rank the Ambrosian and Mozarabic liturgies as full rites in the same sense that the eastern rites are.
That’s pretty much what I said earlier in this thread:
Remember, too, that it is a Rite of the Latin Church and not a Church sui juris on its own.
They are essentially classified as versions of the Latin rite, more like uses. The people and clergy who attend these Masses are all members of the Latin rite, no acknowledgment is made for them today as members of a unique rite or church (as might have been the case many centuries ago). They are considered Latins and nothing else.
That’s one of those “yes, but …” things.
For example, a priest of the of the Ambrosian Rite offers Mass in the Ambrosian Rite unless he is serving in a non-Ambrosian church. And even then, he carries with him the personal faculties to offer Mass in the Ambrosian Rite if he so chooses. (The same applies to the other surviving territorial Rites of the Latin Church as well.) By contrast, a priest of the Roman Rite may
not offer Mass in the Ambrosian (or any other non-Roman) Rite, irrespective of his location.
The proper Rites of the various Orders are somewhat similar, although things have becomes a bit “grey” in the post-conciliar era for all but the Carthusians. The reason for the “greyness” is that, while the traditional usages were never suppressed and still exist “on the books,” the various General Chapters voted (in the late 1960s) to adopt the Novus Ordo as their primary vehicle. But even so, if, e.g., a Dominican were to offer the EF today, he would do so according to the Dominican Rite and not according to the Roman.
Each Catholic Church sui iuris can have one or more liturgies.
The Latin tradition Latin Church sui iuris has several liturgical rites currently used:
Roman Rite Extraordinary form (1962 Missal)
Roman Rite Ordinary form (1970 Missal)
Carthusian Rite (Carthusian 1981 Missal)
Bracarensis Rite
Ambrosian Rite
Anglican Use
Along with the Ambrosian and Bragan, there are two other surviving territorial Rites: the Mozarabic and the Lyonais. (The last was moribund with the advent of the Novus Ordo, but has seen some signs of life in the wake of Summorum Pontificum.)
Along with the Carthusian Rite, there are several other Rites proper to religious orders that survive: the Dominican Rite, the Cistercian Rite, and the Carmelite Rite, albeit that the last one is in a coma. (The Norbertine Rite also survived Trent, but in Trent’s wake, the General Chapter voted to abandon (not suppress) it in favor of the Roman Rite. In theory, at least, it would be possible for another General Chapter of the Order to re-embrace it.)
The Anglican Use is not properly considered a “Rite” at this point. Perhaps that will change when the provisions of Anglicanorum Coetibus are fully implemented.