Hence my need for someone to help me shed my ignorance.
No worries! We’ll worth through it together.
A grounding question for you - how comfortable are you with your understanding of the Latin Church hierarchical structure? It would be easiest perhaps to analogize if that were the case.
Most American-born Latin Rite Catholics I know (including my mom) would tend to think that it stops at the Archdiocesan level. Archdioceses these days tend to be bounded geographically within a specific country. This has helped the Roman Church avoid (or work around) some of the geopolitical issues that have challenged both the Eastern Catholic and Orthodox Churches organizationally, as you’ll hopefully come to appreciate. Larger Archdioceses (those around major cities) are sometimes organized as Metropolitan Archdioceses, and it is customary that the Archbishop also be a Cardinal of the more significant ones. From this point (Archdiocese or Metropolitan Archdiocese), the next level is Rome.
In the East, you may have Patriarchiates and / or Major Archiepiscopal Churches that sometimes can and do cross political borders and divides. In the case of the UGCC example I cited earlier, all global Ukrainian Greek Catholic churches are united under one hierarchical structure (with Archeparchies, Eparchies and Exarchates). Think of these as multinational Archdioceses, if you will. From either of these points, the next level is Rome.
For smaller Eastern Catholic Churches like my Ruthenian Church in the US, while we have close spiritual and traditional ties to our sister Churches in Eastern Europe, we are organized as a separate, nationally based Metropolitan Church (one Archeparchy and three suffragen Eparchies). This also becomes a bit of a source of confusion for some, especially in comparison to the much larger UGCC with which we also share many similarities. Although we all share the same Rite and specific form of Liturgical expression as our sister churches in Eastern Europe, we are organized as separately. In the US, the Ruthenian Church is a separate
sui juris (self governing) Metropolitan Church, whereas our Eastern European counterparts are separately organized. In the case of a
sui juris Metropolitan Church like that of the Ruthenians in the US, the next level is Rome.
So, if you are standing in front of a Patriarch, a Major Archbishop, a Metropolitan Archbishop and an Eparch (Bishop) - all of Eastern Byzantine Rites and fully vested, how do you tell who’s more senior? Look at their individual omophor.
5 bars = Patriarch or Major Archbishop (so you’d have to do some more homework, but each can be called Your Beatitude)
4 bars = Metropolitan Archbishop
3 bars = Eparchial Bishop
Hope this helps, and I’m sure Vico will add (more coherently than I in all likelihood)!