::The first one lists uses of the term in various books, that in it self doesn't mean that there is a religious tradition distinct from other versions of Christianity. I think what they are getting at is Peter's headship of the Church as opposed to say, James. The other two give hypothesis, not fact- Petrine Christianity is not a religious tradition, it is part and parcel of Pauline Christianity. (Note those sources also mentioned Johnainine Christianity, we don't have that in the list... Jesus founded the religion, Paul codified the religion, Peter Lead the religion.