But "pater" is Greek and the Greek word for "father", as well as the Latin word.
My question would be how did the word "Petros" in Greek, which was "Petrus" in Latin, evolve into "Pietro" in Italian and then "Peter" in English???? Wouldn't theologians want to carry over a person's name from one language to another as close as possible to the original language. It's done with "Cephas" for Peter. Then why not "Petros"???
It is conceivable that as the Roman Church began to view "Petros, Petrus" as the Pater [Father] of their Church, that "Pater" would have been Petrus's title: "Pater Petrus" [Father Peter] -- shortened in time to just the Italian "Pietro" and later "Peter", a name remarkably similar to "Pater" in both spelling and meaning to those in the RCC -- Peter, the Pater of their Church.
In Greece at the temple of Delphi, the main object of worship was "Petra". At the temple were little niches in the walls that held small idols and these were called the "Petrae". The sacred writings that were used in the ceremonies were called "Peteroma", Peter's Book!
The temples themselves were also called "Peters". The one temple at Elis in Greece was known as "Petron"; at Delphi it was called "Petraessa". The temple dedicated to Apollo in Asia was called "Patara" and it's oracle was "Patareus". "Peter" worship is traced back to ancient Mesopotamia at the Tower of Babel and the chief Peter god was none other than Nimrod [Genesis 10:8] of the Babylonian Mysteries. Simon Magus, coming from Samaria was an heir to this mystery religion. It was his ancestors who were brought from Babylon to fill the void left by the departed Israelites [II Kings 17:24-34] in Samaria. They were blending the worship of God and paganism even here! Simon took it to Rome where it flourished!