Funny, "the Church of Peter" was being referenced well into the THIRD century...
The Lord says to Peter: "I say to you," he says, "that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it" [Matt. 16:18]. On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [cf. John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. . . . IF SOMEONE [today] DOES NOT HOLD FAST TO THIS UNITY OF PETER, CAN HE IMAGINE THAT HE STILL HOLDS THE FAITH? IF HE DESERTS THE CHAIR OF PETER UPON WHOM THE CHURCH WAS BUILT, can he still be confident that he is in the Church? (The Unity of the Catholic Church 4, first edition [A.D. 251]).
I know, I know. Word must have traveled fast to North Africa since Cyprian already entitled his work as "The Unity of the Catholic Church".
Let's go back to St. Ignatius' letter to Smyrna. That's 110 A.D.:
"Wheresoever the bishop shall appear, there let the people be, even as where Jesus may be, there is the universal [katholike] Church."
Why in the world would they ever call it that? Peter, according to my Bible, was never in charge of much!