Do the religious figures leading a particular See of Christianity have a specific title(s) for that office? For instance:

Pope Francis, Bishop of Rome, _____ of the (Western) See

Emperor Claudius XI, last _____ of the (Eastern?) See

I've also seen the term "synod"; I'm not sure where that fits in either. 

I'm not looking for other political distinctions.
Typically, in Catholicism, in Eastern or Oriental Orthodoxy, and in the Assyrian Church of the East, the highest-ranking bishops of a particular see are given the title Patriarch. The see itself can also be called a patriarchate.

Note however that this title of patriarch applies, within the Catholic Church, not only to the pope as head of the whole Catholic Church but also to the heads of the various Catholic Churches sui iuris, such as the Melkite and Maronite Catholic Churches. I'm not aware of any title that applies to (for example) only the pope within the Catholic Church and also to the heads of all other Orthodox and Eastern churches.