Ignatius of Antioch first to use Catholic.
Those of the Jewish religion were the first to proclaim a new/different order, if you will. The Jewish faith was closed. One pretty much had to be a Jew to participate. Non-Jews were not particularly welcome.
On the other hand, this new order was open and welcoming to all--in the word of the day, catholic. Catholic began as an adjective, not a noun. It later developed into a noun. We can compare this to homeless. Homeless began as an adjective--homeless people--but it has since come to be used as a noun. The homeless.
If I am keeping my players straight, then like you, I believe the catholic church existed since Pentecost. It was drawing people to it.
Today, this can be compared to one TV network presenting a new hit. Before long, other TV networks are going to try to draw viewers away from that hit with a similar offering of their own. That does not mean the original hit won't continue to maintain its own identity. The Early Church insisted on maintaining its original identity. Other "copycats" eventually dissipated.
Or, at least that is how I see it.