And Cyprian lost out in the dispute.
.... In the west, the universal jurisdiction and supremacy if the Pope was virtually unquestioned. Where as in the east, it Ws a lot more complicated. There were some who believes in the primacy of honour alone ( majority) and some who believed as those in the west did.
Absolute nonsense, on two counts. First of all, St. Cyprian of Carthage, whom you are so fond of quoting, once told the Bishop of Rome in no uncertain terms to keep out of a local North African dispute. There are other early examples of Western churches not following Rome's lead.
The Eastern Churches (both Chalcedonian and Non-Chalcedonian) think otherwise.
As Augustine recounts the tale of Cyprian and heretical baptism, he points out how Cyprian later submitted to the authority of the Church:
"Seek counsel from the blessed Cyprian himself. See how much he considered to depend upon the blessing of unity, from which he did not sever himself to avoid the communion of those who disagreed with him; how, though he considered that those who were baptized outside the communion of the Church had no true baptism, he was yet willing to believe that, by simple admission into the Church, they might, merely in virtue of the bond of unity, be admitted to a share in pardon. For thus he solved the question which he proposed to himself in writing as follows to Jubaianus:
"But some will say, 'What then will become of those who, in times past, coming to the Church from heresy, were admitted without baptism?' Lord is able of His mercy to grant pardon, and not to sever from the gifts of His Church those who, being out of simplicity admitted to the Church, have in the Church fallen asleep."
(Augustine, On Baptism, II.18)