Oh, I wasn't looking for a particular answer, just an answer. Your answer was pretty much what I suspected from the flavor of Arcanis, but it pretty different from other games. Your reasoning is totally sound in terms of the nature of churches. I just wasn't sure how differently a corpus cant cast by one faith was from that cast by another. So if you worshipped one god, and switched to another, then you could still cast "Bleed" by just switching the names, or if the entire spell is different between faiths. (Especially since it doesn't really appear that one needs to belong to a specific church to get magic from a god- the gods don't appear that picky - or else there wouldn't be all these cults and schisms that all worship the same god and continue to get power from them - its seems to be all about the faith of the caster, not the acceptance of the god)
I was also considering if an Elorii could worship Nier (I know he would be probably be hunted by other Elorii) as containing "all that remains" of his own fire elemental lord, or as an aspect of him (or containing an aspect of him). But then what would happen if his fire elemental lord returned? Not really a switch of faith, but a change in understanding. IF he had been a priest of nier, would he be unable to switch to the elemental lord of fire?
Sounds like it, which isn't really too much of a surprise.
Thanks for the answer!