The Catholic Church's stance on the papacy of Rome is the following: the Pope of Rome has universal jurisdiction over the whole Church, as the successor of Peter, who is the rock upon which the Church is built and who alone received the Keys of Heaven and made authoritative declarations among the apostles. He can never officially proclaim something false to be the truth, and as such can proclaim dogma infallibly on his own, under certain circumstances.

The Eastern Orthodox Church's stance on the papacy of Rome is the following: the Pope of Rome has jurisdiction over the Church of Rome, and nothing more. However, while he is equal to every other bishop, he is given a high place of honor due to being the successor of Peter and due to ruling over a Church that has held onto the orthodox faith: as such, he may solve disputes and appoint bishops outside of his own jurisdiction, if required.

I am curious to know, which view is closer to how the papacy was understood and exercised during the first centuries of the Church, as well as the 1st millenium?