They’re “socialist” in a purely idealist sense.

I mean they are kinda supposed to be - this is how their leader Nicole speaks/quotes about her in Fallout 1

They claim to want peace, like us. But anyone who disagrees with them just disappears. Their Dark God is not what he seems, believe me.

We never give up hope. We never will, even though most live by the gun these days. And we are concerned with the Children of the Cathedral...

We don't worship so much as follow a set of principles. We want to bring peace back to this wasteland. The world tends toward destruction, so we try to make a difference

After my parents were killed by a marauders, it was time the killing stopped. It's our goal to make this land the way it used to be, before the war.

Nicole is our leader. She's compelled by insight and spirituality to bring peace to a world broken by war.

See the absence of any socialist big words? They are much more christian moralistic in nature.