Why does it matter which happened first? God is the one who is in control, not Peter. He set it in motion by giving Peter the vision.
The second person of the Blessed Trinity delegated certain authority to all of the apostles and delegated certain authority to Simon Peter. He also charged the apostles with certain responsibilities and charged Simon Peter with particular ones.
He set it in motion by giving Peter the vision.
Yes he did.
We (collectively) are the Body of Christ (1 Cor 12). Christ is the head. Peter had a certain function in the body as did Paul. The revelation to make this fundamental change needed to come to Peter, in his function in the Body. It needed then to be communicated to Paul and the rest of the members of the Body.
When the informal guidance that Peter provided started to be misinterpreted, they formalized that guidance through a Council of the Church...where it was codified (Acts 15:6 and following). As the result of this Council, they issued a formal communique: "The brethren, both the apostles and the elders, to the brethren who are of the Gentiles in Antioch and Syria and Cili'cia, greeting. Since we have heard that some persons from us have troubled you with words, unsettling your minds, although we gave them no instructions, it has seemed good to us, having come to one accord, to choose men and send them to you with our beloved Barnabas and Paul, men who have risked their lives for the sake of our Lord Jesus Christ. We have therefore sent Judas and Silas, who themselves will tell you the same things by word of mouth. For it has seemed good to the Holy Spirit and to us to lay upon you no greater burden than these necessary things: that you abstain from what has been sacrificed to idols and from blood and from what is strangled and from unchastity. If you keep yourselves from these, you will do well. Farewell."(Acts 15:23-29)
It matters which happened first because we want to recount accurately what happened in the early Church. Because an error in that area can lead to errors in other areas, as well. Including doctrinal error.