Well, yes and no.... "Mystery Cult" can be a very specific label for a certain type of Greco-Roman religious practice. Christianity does not fit that definition. It can also be used in a wider sense of a religion with an emphasis on mystical experience, which Christianity traditionally is (at least in theory).
I think the point of confusion is that most church attendees are not very far along in their initiatory lives. There is a big difference between knowing that Holy Communion is sharing the body and blood of christ in union with fellow believers and actually experiencing it as that. If you are just eating some flat bread and doing a shot of grape juice or wine, it ain't much of an experience. But when you enter the mystery open to actual mystical experience, its genuinely a powerful experience.
Modern 'low church' and mainstream protestant practice tends to veer away from the mystical, sadly. But high church rites and pentecostals very much are still mysteries.
Anyway, back to the game.... Obviously different cults will have different views on things. That's pretty explicit in the rules already. But none of them can 'just' share their lore with other people. It doesn't work. No matter how much someone writes down the 'secrets' of Enigmatic Wisdom, no one is going to learn it without effectively initiating and changing their worldview.
Even practical matters like advanced alchemy are more than just information you could get from a book. Hermetic magical practice (in the historical sense) is about changing the self to conform to higher and higher levels of reality. Bonisagus and his House has already distilled the "mere information" gleaned from many Mysteries.
The difference between Hermetic shapeshifting and the Heartbeast or between Hermetic enchanting and Verditius magic isn't just "lack of knowledge." Similiarly, beyond hermetic and vulgar alchemy, you are not just taking graduate level alchemy classes, you are involved in a mystical process of changing oneself and how one sees the world such that you now can interact with substances on a higher, more philosophical level. You are fundamentally changing the way your magic works on the world. Its not the same as getting more ranks of Muto or Magic Theory.
So cults should definitely have many different views on how their mystery should be shared. But even the most open won't be able to just say "oh yes, the formula is e=mc2. Now you know."