I wonder how many generations it took for this proto-church to decide it was “the one true church” and start the process of evolving into what we call the Catholic Church today?
It would be a very interesting history project, especially since a lot of history has been so rewritten.
It started proclaiming this bigtime when Reformation leaders successfully challenged its authority.
At that point they were desperate to try to reclaim what they had lost so began to condemn all other believers as heretics.
That practice continues.
Around the year A.D. 107, a bishop, St. Ignatius of Antioch in the Near East, was arrested, brought to Rome by armed guards and eventually martyred there in the arena. In a farewell letter which this early bishop and martyr wrote to his fellow Christians in Smyrna (today Izmir in modern Turkey), he made the first written mention in history of "the Catholic Church." He wrote, "Where the bishop is present, there is the Catholic Church" (To the Smyrnaeans 8:2). Thus, the second century of Christianity had scarcely begun when the name of the Catholic Church was already in use.
Thereafter, mention of the name became more and more frequent in the written record. It appears in the oldest written account we possess outside the New Testament of the martyrdom of a Christian for his faith, the "Martyrdom of St. Polycarp," bishop of the same Church of Smyrna to which St. Ignatius of Antioch had written. St. Polycarp was martyred around 155, and the account of his sufferings dates back to that time. The narrator informs us that in his final prayers before giving up his life for Christ, St. Polycarp "remembered all who had met with him at any time, both small and great, both those with and those without renown, and the whole Catholic Church throughout the world."
We know that St. Polycarp, at the time of his death in 155, had been a Christian for 86 years. He could not, therefore, have been born much later than 69 or 70. Yet it appears to have been a normal part of the vocabulary of a man of this era to be able to speak of "the whole Catholic Church throughout the world."