Ignore anything season 7 said about Slayer & Watcher origins, because I am. The setup that follows is the backstory I've invented for my own nefarious purposes. If I vary from Buffy canon, it's because I want to. Nyah.
In this story, Slayers are chosen of God (aka 3 particular Powers who decided to gang up a long time ago). Christianity is the only religion with holy symbols that are effective against vampires, because it's the one whose backers are annoyed with vamps. Every religion is real, in the sense that it has a Power behind it. (Ethan worships Janus because Janus is a real Power, who rewards Ethan.) Watcher magic is Christian magic.
The deal is that the Council wasn't always the vamp-fighting Slayer-helping organization. The Council as shown was founded by Hank 8 shortly after the dissolution of the monasteries, when somebody cleverly pointed out to him that he still needed somebody to deal with the vampire problem London has had for its entire existence. Before the council, there was a Roman Catholic organization, founded in the dark ages. It still exists, though much diminished because it lost the political squabble with the CoE branch. (England's political ascendence mattered.)
What is the name of that organization?
In "Gas-ring Alchemy", I named the Watcher school St Gregory's, for no good reason whatsoever. But I might work with that. Gregory was an early pope, the first monk to become pope. 6th century. Bleah.
What do these organizations get named after? People, places. Argh! I just need some kind of decision here.
With such issues I lay waste my days.