The first two "rocks", by the way, are problematic. Where do you see "rock" in the Hebrew or in the Septuagint in Dut 32 or 2 Sam 22?
It is not incorrect, however, to see rock as a type of Christ in light of the book of Daniel and 1 Cor 10. Therefore, when Christ names Simon Peter, he is grafting Peter onto Himself, and it is because of the revelation of the Father, correct.
Again, the renaming eposide should be seen together with the giving of the keys, the charge to feed the sheep and the promise of Christ's prayer so that Peter coverts his brethren.
For Jewish Biblical teaching on the subject - and the references to Deuteronomy and 2 Samuel and others, see here:
Another common title of YHWH is "the Rock" (Deuteronomy 32:4,18, 1, 7; I Samuel 2:2; II Samuel 22:32; Isaiah 44:8; Psalm 18:32), thus comparing Him to a high crag on which one finds refuge and safety.