There is also a second form of the mass which was the main form used from not long after the time of the council of Trent until not long after the second Vatican council.
On this point I fear you are quite incorrect - the pre-Tridentine uses such as the Dominican, Carmelite, Carthusian, Norbertine, Sarum, York, Bragan, and Lyonaise uses are all nearly identical to the Tridentine mass, with only minor differences in wording, rubrics and the liturgical calender. If you don’t believe me, read the missals. There are English translations of the Sarum and York missals that I have in my liturgical library (and I believe they exist of the Hereford and Durham uses as well), as well as of the old Dominican and Carthusian masses. And you can also find online the Latin originals, which of course I also have, and have read.
The Tridentine mass was not in any respect a major change from the plethora of regional uses it was intended to consolidate (which I regret, naturally - I wish they hadn’t done that, since even though those places which could show their use had been around for more than 200 years could keep it, not everyone bothered, and as for the four English uses, since by the time of Trent Catholicism was officially illegal in England (which was horrible) and was secretly practiced underground by composers such as Byrd and Tallis, well, they simply ceased to be practiced, although the missals survived along with enough other material so that the Anglo Catholics and Western Rite Orthodox were able to revive them.
But at any rate the idea that Tridentine marked a major departure is simply wrong - it was basically the Roman Rite as it existed at the time of the Council of Trent. One other unfortunate decision of the Council of Trent was to wage a war on the Rood Screens, although they did not fire the first shot - this had been done by the Franciscan and Dominican friars. If it had remained limited to them, it would have been fine, just as how the unusual liturgy celebrated by New Skete Monastery is not a problem for the Orthodox unless it starts being adopted by parishes, which is not going to happen. At any rate, the loss of most of the Rood Screens in Europe caused a further rupture with the ancient tradition which survives in the form of the Byzantine, Coptic and Ethiopian iconostasis (some Syriac Orthodox parishes have an iconostasis, but usually the icons are displayed not so much on a stasis but on columns or otherwise in a manner adjacent), the Armenian Bema, and the curtain used by the Armenians (together with the bema), most Syriac Orthodox parishes, and the parishes of the Church of the East. Fortunately, some survived, ironically, in Anglican churches, albeit with iconoclastic defacement, and some of these were restored, and new ones built in the 19th century.
Now, there is an alternate form of the traditional Latin mass, but it is a different rite, or family of rites, specifically the Gallican Rite and its relatives, the Ambrosian, Beneventan, and Mozarabic Rites. These follow a different ordo from the Roman Rite.
Of these, the Gallican became extinct after Charlemagne ordered it replaced with the Roman Rite, but we have some surviving manuscripts, and the same is true of the Beneventan rite used in Southern Italy (Campania, I think).
However, the Ambrosian rite is still celebrated by over a million people in the Archdiocese of Milan, and likewise the Mozarabic Rite also survives, but unfortunately it went from having seven parishes in the 19th century to having none today - it is preserved only in a chapel in the Cathedral of Toledo, and in a nearby monastery, and in some of the marriage rites used in Mexico, but it was celebrated by Pope John Paul II.
The interesting thing about the old Gallican rites is that they, along with the East Syriac Rite used by the Assyrian Church of the East, and to a lesser extent the Syriac Orthodox liturgy, where this was once common, but now usually does not happen, are the only ancient rites where the Old Testament was read in the
Synaxis (what is called “the Liturgy of the Word” in modern RC parlance).
By the way, the term used for the Introductory Rites, well, parts of them, is the
Prothesis,
Proskomide or Liturgy of Preparation or
Preparatory Rites, the correct term for the Liturgy of the Word is the
Synaxis, or Liturgy of the Catechumens, and the correct term for the rest is the Liturgy of the Faithful, which contains the
Anaphora, which the Novus Ordo insists upon calling “the Eucharistic Prayer”, a term I find very frustrating because the
entire liturgy is a Eucharistic prayer - that’s what the mass is.
Speaking of which, the Roman Canon (“Eucharistic Prayer 1” in the Novus Ordo) is the only anaphora known to have been celebrated in the Roman Rite. There is no evidence the anaphora contained in the Apostolic Tradition of Hippolytus was ever used in Rome, but it was used in Antioch, and the Antiochene liturgy was the most widely used in the ancient world, which is why this liturgy is used almost word for word, albeit in its full form, not just the parts for the priests (one of the two major mistakes Bugnini made in composing Eucharistic Prayer 2 was to assume that the text from St. Hippolytus was the entire anaphora, which it wasn’t - ancient liturgical books unlike the Roman Missal were written for specific users, and contained only the text relevant to those celebrants - the Roman Missal introduced the concept of having everything in one book, and this was followed by the adoption in the Roman Rite of the practice, which still survives in the Latin Mass, of the priest repeating
everything in his prayers, in addition to having his own prayers to say, such as the Secret and most of the Canon (the other mistake Bugnini made was to then modify the text so that the Epiklesis would precede rather than follow the Words of Institution, which he also did to the adaptation of the Egyptian form of the Divine Liturgy of St. Basil, known to liturgists as EgBAS).
At any rate, what St, Hippolytus provided was the ancient liturgy in Antioch, known as the Anaphora of the Apostles, which is still in use in the Ethiopian Tewahedo Orthodox Church, which uses an Antiochene liturgy that was taught to the Ethiopians by the Seven Syrian Sages, who were clergy from Antioch who came to assist the country in its mass conversion, which is interesting, since in the fourth century Egypt predominantly used the Alexandrian liturgy, known as the Divine Liturgy of St. Mark or the Divine Liturgy of St. Cyril among the Copts (because he translated it into Coptic), which is the oldest attested liturgy and is believed to be one of the two oldest in existence, along with the East Syriac liturgy of Saints Addai and Mari. But even then Alexandria was introducing Antiochene liturgies, in the form of EgBAS, and the Divine Liturgy of St. Gregory the Theologian, which my friend
@Andrewn asked me about a few months before I became spectacularly ill in the spring.
Variant forms of this liturgy are also used by the Syriac Orthodox and the Eastern Orthodox (Divine Liturgy of St. John Chrysostom, which makes sense because St. Chrysostom was a priest in Antioch before being consecrated as Patriarch of Constantinople) and by the Maronite Catholics.
For more information on this, see
the Oxford History of Christian Worship, Essays in Early Eastern Eucharistic Prayers, Prayers of the Eucharist: Early and Reformed, Context and Text: A Method for Liturgical Theology, Issues in Eucharistic Praying: East and West, Do This In Remembrance of Me, Liturgical Reform After Vatican II: The Impact on Eastern Orthodoxy (spoiler: there is none), and several other books, plus various pre-Tridentine missals which I have no doubt that with the use of Google you can find as easily as I have.