Peter and Peter's Confession--one and the same for the early Church Fathers?

The Myth:

Robert Sungenis (Roman Catholic apologist):

"When you look at all the early Fathers, for instance, they never distinguish Peter the person from Peter's confession of faith; they were one and the same." (35:45 mark)

The Reality:

John Chrysostom:

He did not say, “upon Peter” for it is not upon the man, but upon his own faith that the Church is built. And what is this faith? “You are the Christ, the Son of the Living God.” (Migne, 52.806.75-807.1)

Therefore He added this, “And I say unto thee, Thou art Peter, and upon this rock will I build my Church;” that is, on the faith of his confession. Hereby He signifies that many were now on the point of believing, and raises his spirit, and makes him a shepherd. “And the gates of Hell shall not prevail against it.” And if not against it, much more not against me. So be not troubled because thou art shortly to hear that I shall be betrayed and crucified. (Homily LIV)

Augustine

Why have I wanted to make this little introduction? In order to suggest to you that in Peter the Church is to be recognised. Christ, you see, built his Church not on a man but on Peter’s confession. (Sermon 229)