The Roman Catholic Church is headed by the pope, on that I'll agree.
Cordially,
Just for the record, it is the Catholic Church. It is headed by the pope, who is in Rome, of which he is a bishop, or it can be also said that he is in the Vatican. The Church recognizes several "rites", that is liturgical and ecclesiological systems. They all share the same Catholic theology, but differ in how the liturgy is conducted, how the sacraments are administered and in the disciplines of their priests. Most people are only familiar with one, Roman or Latin rite that prevails in Western Europe and the Americas. But in Slav lands and in the Middle East there are about twenty small rites that are also Catholic, even though in appearance they resemble the Orthodox, and they admit married men into priesthood. There is also a rite or two that is Latin in appearance but is distinct and is kept by some monastic orders. The pope belongs to the Latin rite and he administers the Latin rite more closely than the Eastern rites. Catholic laity is free to take communion and generally fully participate in any rite and in fact it would be good if more Catholics visited Eastern rite churches that exist in larger cities, just to broaden their liturgical experience and avoid liturgical blandness that plagues many Latin rite parishes after Vatican II.
It is therefore a bit imprecise to call the Church Roman Catholic, unless a specific reference to the Roman rite is implied.
Thank you for the stimulating discussion of the early papacy, I enjoyed it very much.