I believe the various cults generally use the same forms of magic that are already commonly used by other people and the player. Until shown otherwise, it's safe to say pretty much all magic by "regular people" will basically boil down to some form of arcanima, astrology, conjury, or thaumaturgy. There are specializations and corruptions we've seen, however.
The only one that comes to mind are the Lambs of Dalamud, who tend to use a mix of Thaumaturgy and void magic.
The only one that comes to mind are the Lambs of Dalamud, who tend to use a mix of Thaumaturgy and void magic.