"The twenty-five years' episcopate of Peter at Rome is evidently due to the statement of Justin Martyr regarding the labors of Simon Magus at Rome (see above), combined with the tradition of Peter's residence in the same city, especially as it would seem that the Roman Church had actually been formed early in the reign of Claudius through the indirect influence of the Petrine Christianity of Palestine."
Note that the encyclopedia refers to Peter's residence as "tradition" not fact, but "as it would seem", and the phrase "the indirect influence" meaning what? If Peter had been involved in its formation it would have been "direct influence", but the encyclopedia says "indirect" meaning that Peter was not there personally, but what was Petrine Christianity in Palestine made its way to Rome indirectly.
Why don't you finish the sentence:
Why don't you finish the page: "All this giving rise to ... the legend of his twenty-five year residence in Rome"
And this:
"it was not until the middle of the third century that Peter was definitely claimed as bishop of Rome".
So here we have the answer to our question of when Peter became bishop of Rome: Peter became bishop of Rome in the middle of the third century by virtue of the pen of Cyprian and embellished upon a short time later by Eusebius.
Please note yet another paragraph in the article:
In opposition to this distinct and unanimous testimony of early Christendom, some few Protestant historians have attempted in recent times to set aside the residence and death of Peter at Rome as legendary. These attempts have resulted in complete failure. It was asserted that the tradition concerning Peter's residence in Rome first originated in Ebionite circles, and formed part of the Legend of Simon the Magician, in which Paul is opposed by Peter as a false Apostle under Simon; just as this fight was transplanted to Rome, 80 also sprang up at an early date the legend of Peter's activity in that capital (thus in Baur, "Paulus", 2nd ed., 245 sqq., followed by Hase and especially Lipsius, "Die quellen der römischen Petrussage", Kiel, 1872). But this hypothesis is proved fundamentally untenable by the whole character and purely local importance of Ebionitism, and is directly refuted by the above genuine and entirely independent testimonies, which are at least as ancient. It has moreover been now entirely abandoned by serious Protestant historians (cf., e.g., Harnack's remarks in "Gesch. der altchristl. Literatur", II, i, 244, n. 2). A more recent attempt was made by Erbes (Zeitschr. fur Kirchengesch., 1901, pp. 1 sqq., 161 sqq.) to demonstrate that St. Peter was martyred at Jerusalem. He appeals to the apocryphal Acts of St. Peter, in which two Romans, Albinus and Agrippa, are mentioned as persecutors of the Apostles. These he identifies with the Albinus, Procurator of Judaea, and successor of Festus and Agrippa II, Prince of Galilee, and thence conciudes that Peter was condemned to death and sacrificed by this procurator at Jerusalem. The untenableness of this hypothesis becomes immediately apparent from the mere fact that our earliest definite testimony concerning Peter's death in Rome far antedates the apocryphal Acts; besides, never throughout the whole range of Christian antiquity has any city other than Rome been designated the place of martyrdom of Sts. Peter and Paul.
I could easily repeat the section I quoted before, but I'll not insult the other readers' intelligence through repetitiveness.
As to the Schaff article, even this non-Catholic source states as its closing sentences for this section: It seems most probable, on the whole, that Peter died a martyr's death in Rome toward the close of Nero's reign, some time after the cessation of the general persecution. Absolute certainty is, however, unattainable.
Your statement,