So no matter what rite we are all Roman Catholic?
Before Vatican II, but generally after Vatican I, the normal mode was
“Roman Catholic Church” “___ Rite Parish” or “___ Rite Priest” with the Eastern Churches in Communion with Rome generally subsumed to be subject to Rome.
In many places, Catholics of the Eastern Rites were in fact part of Roman Dioceses, with their priests incardinated in the Roman Church’s Dioceses. The opposite was also true in some areas - in Ethiopia, all the Catholics were under Ethiopian Rite bishops.
Problems in the US, involving both Eastern Rite and Dominican Rite priests being forbidden to use their own rite’s liturgies by the 3rd Plenary Council of Baltimore, were finally brought to a head in 1908, when the prohibition on non-Roman liturgies in the US was overturned by Rome, and rome specified that Catholics of all Rites have the right to worship according to their native Rite. The US bishops then generally refused to incardinate priests from other rites, even if they were celibate, so a few years later (and in the continued wake of a Ruthenian schism), the parallel exarchates were established for the Ukrainians, Ruthenians, Chaldeans and Melkites.
Prior to that point, the parishes were “Roman Catholic” in the sense of being under Roman Bishops. Some old parishes still have signage reflecting that origin; others adopted it as a way of saying “Yes, Romans can fulfil their obligation here”…
As an aside, the Roman Rite bishops then imposed upon Rome to ban married priests from service in the US. Rome imposed Cum Data Fuerit, barring married men from Ordination in the US and incardination into dioceses in the US… the Ukrainians’ Archeparchy was in Canada, and so was technically not “in the US” even tho’ it had parishes in the US, and so they’ve maintained married clergy the whole time… The Ruthenians, however, suffered a SECOND schism over Cum Data Fuerit…
Vatican II decided the “Church-ness” of the Eastern Churches needed to return, and declared that the diversity of Rites was not only of value, but essential to the health of the Church, and that the post-conciliar documents of both Trent and another council calling for eventual abolition of the Eastern Churches in favor of the Roman Rite were in fact in error. It also called for the Eastern Churches to return to functioning as Churches, rather than mere dioceses of different rite, complete with internal autonomy, conciliarism, and a return to traditional liturgies.
Vatican II also had a committee that deemed the term “Rite” was overused, and confusingly so, and decided to count 6 “Rites”, with various minor rites… and put the various churches into one each. (But then those Ethiopians still had both Alexandrian Rite and Roman Rite, with both Ethiopian and Eritrean uses of the Alexandrian Rite, and the Roman Church has both Roman Rite, the several hybrid Gallo-Roman Rites, several monastic Rites, and portions of the Byzantine Rite … it’s not as clear cut…)
Anyway, Since V II, the term Rite has ceased to canonically mean a Church in Communion with Rome, and only to mean a Major Rite (Roman, Alexandrian, Chaldean, Syrian, Armenian, Byzantine), a western sub-rite (Mozarabic, Ambrosian, Bragan, Dominican, Carmelite, Carthusian, Norbertine, Dalmatian Rites), or a specific chunk of Ritual with a particular Purpose (Rite of Acceptance, Rite of Baptism, Rite of Confirmation, Rite of Ordination, Rite of Sprinkling, etc). All of which were also meanings of Rite pre-V II, but by removal of “Church Sui Iuris in Communion with Rome” from the list, it’s reduced the practical confusion somewhat. There’s been a move, in english at least, to replace “rite” as in a specific chunk of ritual with “liturgy” “ritual” or “ceremony” according to specific elements (such as Liturgy of the Word replacing Rite of Readings) but I am not certain what level that’s coming from, whether it’s approved, and whether it will stick.