Joab,
First, there is no Eastern Catholic Church. There are a variety of Eastern and Oriental Catholic Churches.
What separates the EC from the EO is the intent on being in communion with Rome. Mind you, many of these churches existed and came into union far before the definitions of infallibility, IC, and such existed. As such, these were non-issues when communing with Rome. The most prevalent example would be the Melkites who shared union with both Rome and Constantinople for a dramatically long period of time after the schism, again, lacking the developments found in the West. This is why the Melkites in particular will be very fervent (rightly so) defending their lack of original sin, purgatory, IC as defined by being free from original sin (which again they don’t have), etc. I would encourage you to look up the words of the Melkite Patriarchs to hear their views of their office versus Rome’s; you will be surprised.
What separates the OC from the EO is tradition and history. Those liturgical and praxis qualities found in the Oriental Orthodox are what the Oriental Catholics share. To me, this is a non issue, from my personal experience with Oriental Orthodox and Eastern Orthodox, their degree of separation is not noticeable beyond their practice, however, I come from a place where EO, EC, OC, OO commune amongst each other. Again, the issue that would separate the Orientals from the Easterners would be the original intent in communing with Rome.
I don’t subscribe to the Western debates between Oriental and Byzantine theology, from pastoral counsel and personal experience, these issues tend to be resolved, and near fruition at the ecclesiastical level. This would also involve Latin commentary on Syriac tradition, which more often than not is flagrantly misrepresented to defend things such as the IC and universal jurisdiction.
Peace and God Bless.
