Yes I guess it is confusing for you as well as others who do not know or fully understand how, what you call the Byzantine Rite papal Catholic Church' came into existence in Eastern Europe in late 16th and early 17th century. As the old saying goes....'It's not what's on the outside that counts, it's what on the inside!' This situation is a perfect example.
Here's a brief rundown -
Prior to 1596 the people who are now called Greek or Byzantine Catholics in Eastern Europe were all members of the Eastern Orthodox Catholic Church. When these lands were taken over by Roman Catholic empires (Austro-Hungarian, Polish, Lithuanian, etc, rule) these RC empires tried to force the people into the RCC. The Protestant reformation had begun & the RCC was losing millions of souls. The people themselves tough peasants were very devout in their Orthodox types of worship and identity. The now RC government knew they would never willingly give up their Orthodox identity or beliefs. They also realized, that because it was the 16th & early 17th century, the people could neither read nor write. They based everything on what they saw and heard. To them...AS LONG AS EVERYTHING LOOKED THE SAME AND SOUNDED THE SAME...IT WAS THE SAME! The plan was that over time the RCC would start the process of Latinization with each new generation. Though it may take a century or more, the time would come when the new church under Rome would become fully latinized and no different than its counterpart in Rome. So when the Unia was signed evrything stayed the same. With the exception the Popes name was commerated in the main Cathedral but the local Bishop was still mentioned everywhere else. So as far as the people were concerned one Sunday they went to Liturgy and were Orthodox. The next Sunday they went to Liturgy at the same parish but were now papal Catholics and were none the wiser. In the main Cathedral where the Pope was commerated and the people questioned it they were told the Pope had become Orthodox! Because of this, some of them still claim they are 'Orthodox In Communion With Rome!' which is an oxymoron!
My grandparents came to this country knowing what had happened but there were others who came from areas where they had no idea they were no longer Orthodox until they came here. I personally consider this as a cancer perpetuated by the RCC on innocent people. And all for the lust for power and glory. No matter how much the RCC tries to justify what it did, it was no coincidence that it happened around the same time Rome was losing millions of its people to Protestantism. It has caused many rifts within families (those that returned to Orthodoxy vs those that remained). My grandfather and his brother lived in the same small town and were no longer speaking to each other after our Orthodox Church was built.
Remember, a persons faith or religious identity is not based on how they worship but what they believe (contained in the doctrines & dogmas of their church). If you believe in what the RCC teaches then you are a Roman Catholic or papal Catholic. If you believe what the OCC teaches and believes then you are an Orthodox Catholic.
When I hear people here and elsewhere claim that they believe all that the Orthodox Catholic Church teaches but are knowingly and willingly in communion with the pope and accept him as the highest earthly authority in their Church I can only shake my head in bewilderment. Because to me, its like saying that they are knowingly & willingly under the authority of a heretical bishop!
This is put very well. I would add only that it was not simply the rulers (the King of Poland, etc.) and Rome who desired Catholicism within his realm, but many of the (we would say misguided) Orthodox Bishops in Eastern Europe also initially desired union for purposes of religious protection and advancement. This was for several reasons: the Patriarch of Constantinople, to whom the Metrpolitan of Kyiv and Halych had been subject, had been greatly weakened after the fall of Constantinople to the Turks in 1453. It was not easy to communicate about religious matters, as the Patriarch was greatly restricted in what he could do, and was surrounded by a hostile regime. They were not interested in switching allegiance to the Patriarch of Moscow, inasmuch as they were under the political protection of the Polish sovereign, and it was not desirable to be ecclesiastically subject to a patriarch in a rival state. Thus, many Orthodox bishops, having come under the jurisdiction of the Polish (Catholic) State, were more amenable to seeking the ecclesiastical protection of the Bishop of Rome than they might otherwise have been. Thus the Union of Brest in 1596. Educated non-clerical people, especially the Orthodox groups in the cities, rejected (in some cases violently) the Union. Some bishops soon after repudiated it, resulting in a messy century-long situation in which some cities had both a Greek Catholic and an Orthodox bishop. Within a century, their successors accepted the Union. You speak about the Byzantine Catholic Church. It did not stem from the Union of Brest, but from a similar union adopted in 1646, called the Union of Uzhhorod, covering the lands subject to the Eparchy of Mukachevo. If you Google, you can find (in English) the terms of the Union of Brest, which were primarily liturgical.
The people in the villages really saw little change in their churches until after the Synod of Zamosc in 1720 when the Greek Catholic liturgy started to become standardized. Prior to that, the priests were using old Orthodox service books which had been "retro-fitted." Many villages near the border with Russia would call in an Orthodox priest if their Greek Catholic priest had died. This was problematic to the Catholic Church and for the government, which didn't want Russian influence in the parishes. The Greek Catholic Church began a stringent campaign to educate its priests, and it became noted for its sermons. This was so that it could impress upon the people that they were praying for the Pope, who for many years was not included in the prayers in some villages. Latinizations began to accrue.
After the Union, the Greek Catholic Church extended throughout the Polish lands, from Kyiv into what was western Galicia, approaching Krakow, Poland. As Russia acquired more and more territory from the weak and dying Polish-Lithuanian Commonwealth, however, it fought to bring the people back to Orthodoxy. If one reads anything of the life of Catherine the Great, one comes to the conclusion that this was more a political calculation than anything else, just as the Union of Brest had been. Some parishioners returned very willingly to Orthodoxy, and others were compelled to give up their Greek Catholicism. Because of political efforts, the entire region under the Russian partition of Poland reconverted to Orthodoxy in three waves, in 1796, 1839, and 1875.
Church history is messy and politics created some dirty events on both sides, sadly. Modern freedom of religion simply did not exist in the age of "cuius regio, eius religio."
Greek Catholics from both the Union of Brest lands (mostly north of the Carpathians) and the Union of Uzhhorod lands (south of the Carpathians) came to the USA in the late 19th and early 20th centuries. They spoke varying dialects of an East Slavic tongue, calling themselves Rusnaks, Ruthenians, Carpatho-Russians, and/or Ukrainians, and worked in the mines and factories of the Northeast. Some, influenced by St. Alexis Toth, returned to Orthodoxy soon after arrival, and they are the nucleus of what was called the Orthodox Greek Catholic Church (originally under Russia), which is now the OCA. Others remained in the Greek Catholic Church which, by 1924, had split into the Ukrainian Greek Catholic Church and the Byzantine (or Ruthenian) Catholic Church, based on the competing political sentiments of its members. In this split, those who went with the Byzantine or Ruthenian Catholic Church almost all had their origin in the descendants of people from the Eparchy of Mukachevo, south of the Carpathian Mountains in what is now Slovakia and the Transcarpathian Oblast of Ukraine, while those who went with the UGCC came from north of the Carpathian Mountains in what is now Southeastern Poland and the three western oblasts of Ukraine (L'viv, Ivano-Frankivs'k, and Ternopil). Some came out of these groups into Orthodoxy -- those from the UGCC starting in 1929, mainly because married priests were disallowed and parish control of church property was restricted, and organized the Ukrainian Orthodox Church of the USA , and those from the Byzantine Catholic Church in the 1930s came into the American Carpatho-Russian Orthodox Diocese (ACROD) for some of the same reasons. So you can see why all the liturgies of these churches have common variants, deriving from the Divine Liturgy as employed by (broadly) one particular ethnic group.
But today, the theology is standardized in both camps. The UGCC and BCC are Catholic Churches under the Pope, and the UOC of the USA and ACROD are Orthodox Churches under the jurisdiction of the Patriarch of Constantinople. And, in the motherland, the Metropolitan of Kyiv has been under the jurisdiction of the Patriarch of Moscow since 1686, when Russia acquired the territory of Kyiv and the metropolitanate was transferred from Constantinople's jurisdiction.