The term “Catholic” (from Greek katholikos, “universal”) was used by early Christians to describe the Church’s unity across geographic and cultural boundaries. St. Ignatius of Antioch, writing around 107 AD to the Smyrnaeans (one of the seven churches!), says: “Wherever Jesus Christ is, there is the Catholic Church” (Letter to the Smyrnaeans 8:2). This early use shows that even the churches in Asia were understood as part of the universal Church, united by bishops in apostolic succession.
Peter was NOT a pope. Jesus said on this rock (HIM) the church will be built, not Peter. That would be completely contrary to everything Jesus taught.