What's on the wiki you've linked is a fine basis from which to begin! As they say there, the cult itself changes from region to region, based on local customs and beliefs, so any particular one that Hitagi's mother was involved would could be regional. In that case, it wouldn't impinge on others if they have a different sort of Kaiju Cult theme they wanted to work with in a character's back story. As long as the basic beliefs of the BuenaKai are consistent, specifics you come up with should be fine!

"[...] adherents commonly believe that the end times have arrived, and the Kaiju have come to scour the world's sins away."

That's the most important element to have preserved. I know the wiki also states there appears to be good funding coming in, which makes sense on a regional basis, much like it'd make sense for the larger followings to be within the Pacific Rim region, as it's the one where the Kaiju have had direct literal impact. (Elsewhere it's been more indirect.)