Both names for the enemy are fine, there are different circumstances when one or the other would be preferred. The larp is things like "Church of Satan" but since it's a deliberate deception it's still satanic, just not it the surface sense. But whether we call the enemy the satanists, the children of the devil, the sons of Belial, the luciferians, the antitheists, we are using the same core concept.

(To OP: I haven't found OP's links to be worth clicking so I am not commenting on them without enough precis to know what I'm clicking on and why.)