Check this out:
newadvent.org/cathen/13121a.htm
From what I understand, the Church of Rome is the Roman Catholic Church with the Holy See located in Vatican City in Rome, and the Holy See is the head of the one, holy, catholic, and apostolic Church.
There is in the Roman Catholic Church 23 main rites, 22 Eastern and 1 Western, the Latin Rite. The Latin Rite has “sub-rites” within itself. An Eastern Catholic is not a Latin Rite Catholic but is nevertheless Roman Catholic. Roman in this case obviously is not saying that you have to be from Rome. Rome symbolizes the unity of Christ’s true Church.
Just my two cents…