I hope that the video clarifies some of the issues involved. We could provide other quotations from the church fathers as well. For example:
Augustine: "Why does Pelagius choose to be sunk in that sea whence Peter was rescued by the Rock?" (On the Grace of Christ, Chapter 50)
What then saith Christ? “Thou art Simon, the son of Jonas; thou shalt be called Cephas.” “Thus since thou hast proclaimed my Father, I too name him that begat thee;” all but saying, “As thou art son of Jonas, even so am I of my Father.” Else it were superfluous to say, “Thou art Son of Jonas;” but since he had said, “Son of God,” to point out that He is so Son of God, as the other son of Jonas, of the same substance with Him that begat Him, 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 52 on Matthew, Section 3[second section 3, per this source])
Again in the state of the Law the first revelation which was given to Moses was more excellent, and on this revelation all the other revelations to the prophets were founded. And so, too, in the time of grace the entire faith of the Church is founded on the revelation vouchsafed to the apostles, concerning the faith in one God and three Persons, according to Matthew 16:18, "On this rock," i.e. of thy confession, "I will build My Church."(Summa Theologica, 2nd Part of the 2nd Part, Question 174, Article 6)
Now, Aquinas certainly wasn't a Reformed Presbyterian, but (like the other writers of bygone eras quoted above) he didn't share the view of Matthew 16:18 that we hear from Roman Catholic apologists and the modern Roman Catholic magisterium.