Peter, called Cephas, is - by the very Scripture you cited - identified as “a stone”, not “the Rock”. The Rock is Christ, all men are as sand.
John 1:40-42:
“40 One of the two which heard John speak, and followed him, was Andrew, Simon Peter’s brother.
41 He first findeth his own brother Simon, and saith unto him, We have found the Messias, which is, being interpreted, the Christ.
42 And he brought him to Jesus. And when Jesus beheld him, he said, Thou art Simon the son of Jona: thou shalt be called Cephas, which is by interpretation, A stone.”
That was a decision made by the translator of whatever version you are consulting--who decided to de-emphasize the fact that it was a proper name and make it into a common noun. Here is the passage in the original Greek:
Simwn o uios Iwannou, su klhqhsh Khfas, o ermhneuetai Petros
Simon son of John, you will be called Khfas (= Kephas), which is interpreted, Petros (= Petros).
The capital letters are not part of the original of course, so it's not 100% clear which meaning is meant here, but there is certainly no sound basis for insisting that this *has* to mean "small stone", when it could just as easily be the proper name Peter.
In any case, John is saying that Petros is a translation of the original Cephas, so if you want to make hay out of what Peter's name means, you have to do it in the Aramaic.