The second source is anachronistic since there wasn’t a monepiscopate in Rome until the 2nd century.  The point of the link was that the idea that Peter was bishop of Rome is quite problematic with regards to what the early Church claimed. The ECF’s couldn’t agree on whether there were several bishops reigning at the same time in Rome, whether Peter ordained successors before or after certain dates, whether he ordained anyone, how long he reigned etc. etc.
Epiphanius thought Peter and Paul both reigned as bishop of Rome:
“For the bishops of Rome were, first, Peter and Paul, the apostles themselves who were also bishops—then Linus, then Cletus, then Clement, a contemporary of Peter and Paul whom Paul mentions in the Epistle to the Romans.” The Panarion of Epiphanius of Salamis (Leiden: Brill, 1987), Book I, Section II, 27.6,2, p. 104).
The Liber Pontificalis says Peter reigned as bishop of Rome for 25 years:
“Blessed Peter, the Antiochene, son of John, of the province of Galilee and the town of Bethsaidia, brother of Andrew and chief of the Apostles, first occupied the seat of the bishop in Antiochia for 7 years. This Peter entered the city of Rome when Nero was Caesar and there occupied the seat of the bishop for 25 years, 1 month and 8 days.” - Liber Pontificalis, (New York: Columbia University, 1916), p. 4
The ECF’s aren’t reliable on this.
“In the late 2nd or early 3rd cent. the tradition identified Peter as the first bishop of Rome. This was a natural development once the monarchical episcopate, i.e., government of the local church by a single bishop as distinct from a group of presbyter-bishops, finally emerged in Rome in the mid-2nd cent.” - Kelly, Joseph F. The Concise Dictionary of Early Christianity. Collegeville, MN: Liturgical, 1992. p. 6.
“As for Peter, we have no knowledge at all of when he came to Rome and what he did there before he was martyred. Certainly he was not the original missionary who brought Christianity to Rome (and therefore not the founder of the church of Rome in that sense). There is no serious proof that he was the bishop (or local ecclesiastical officer) of the Roman church --a claim not made till the third century. Most likely he did not spend any major time at Rome before 58 when Paul wrote to the Romans, and so it may have been only in the 60s and relatively shortly before his martyrdom that Peter came to the capital.” - Brown, Raymond Edward., and John P. Meier. Antioch and Rome: New Testament Cradles of Catholic Christianity. New York: Paulist, 1983. p 98.