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