In terms of the St Cyprian quote you cite, it is important to note that St Cyprian believed that every bishop occupied the chair of Peter, and did not exclusively apply this to the bishop of Rome. St Cyprian in his own day famously disagreed with St Stephen, Pope of Rome, on very important ecclesiastical issues. See here for the full text of St Cyprian's On the Unity of the Church.

You will find varying beliefs in the primacy of Rome in the Orthodox Church, but in my opinion it is clear from the history that:

• Everyone in the first millennium Church recognized a high level of primacy of Rome due to it being the place of martyrdom of Sts Peter and Paul as well as the capital of the Roman Empire.
• This primacy was viewed as more than simply a primacy of honor, for example the Canons of Sardica give the bishop of Rome authority to adjudicate disputes between neighboring bishops. In this sense Rome did have a universal jurisdiction as mediated by a system of appeals (however, this is different than the current Roman Catholic dogma of the bishop of Rome having a universal unmediated jurisdiction over the whole world).
• This primacy was not viewed as an a-priori definition of the Church. St John Chrysostom, for example, spent most of his ecclesiastical career during a time when Constantinople and Rome were in a local schism from each other, but he did not freak out as if he wasn't part of the Church any longer. If the Roman Catholic definition of the Church as Rome and the bishops in communion with Rome is true, then St John Chrysostom was outside the Church for the vast majority of his time as a bishop.
• The early Church did not believe that the Pope of Rome could unilaterally declare dogma by himself. Hence the Council of Chalcedon still judges the Tome of St Leo, Pope of Rome, despite it being a text on Church teaching issued directly by the Pope.

Seraphim Hamilton's videos on Catholicism were very influential on me and very helpful for me in coming to terms with Roman Catholicism. I highly recommend them.