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Titles: The most noteworthy of the titles are Papa, Summus Pontifex, Pontifex Maximus, Servus servorum Dei. The title pope (papa) was, as has been stated, at one time employed with far more latitude. In the East it has always been used to designate simple priests. In the Western Church, however, it seems from the beginning to have been restricted to bishops (Tertullian, On Modesty 13). It was apparently in the fourth century that it began to become a distinctive title of the Roman Pontiff. Pope Siricius (d. 398) seems so to use it (Ep. vi in P.L., XIII, 1164), and Ennodius of Pavia (d. 473) employs it still more clearly in this sense in a letter to Pope Symmachus (P.L., LXIII, 69). Yet as late as the seventh century St. Gall (d. 640) addresses Desiderius of Cahors as papa (P.L., LXXXVII, 265). Gregory VII finally prescribed that it should be confined to the successors of Peter (1073-85). Source: The Pope (Catholic Encyclopedia).
Below are quotes from the New Advent web site on the titles Bishop of Rome and Pope:
It is no longer denied by any writer of weight that St. Peter visited Rome and suffered martyrdom there (Harnack, "Chronol.", I, 244, n. 2). Some, however, of those who admit that he taught and suffered in Rome, deny that he was ever bishop of the city (e.g. Lightfoot, "Clement of Rome", II, 501; Harnack, op. cit., I, 703). It is not, however, difficult to show that the fact of his bishopric is so well attested as to be historically certain.
The first witness is St. Clement, a disciple of the Apostles, who, after Linus and Anacletus, succeeded St. Peter as the fourth in the list of popes. In his "Epistle to the Corinthians", written in 95 or 96, he bids them receive back the bishops whom a turbulent faction among them had expelled... The tone of authority which inspires the latter appears so clearly that Lightfoot did not hesitate to speak of it as "the first step towards papal domination" (Clement 1:70). Thus, at the very commencement of church history, before the last survivor of the Apostles had passed away, we find a Bishop of Rome, himself a disciple of St. Peter, intervening in the affairs of another Church and claiming to settle the matter by a decision spoken under the influence of the Holy Spirit. Such a fact admits of one explanation alone. It is that in the days when the Apostolic teaching was yet fresh in men's minds the universal Church recognized in the Bishop of Rome the office of supreme head.
The [second] century gives us the witness of St. Irenaeus — a man who stands in the closest connection with the age of the Apostles, since he was a disciple of St. Polycarp, who had been appointed Bishop of Smyrna by St. John. In his work "Adversus Haereses" (III:3:2) he brings against the Gnostic sects of his day the argument that their doctrines have no support in the Apostolic tradition faithfully preserved by the Churches, which could trace the succession of their bishops back to the Twelve. He proceeds to enumerate the Roman succession from Linus to Eleutherius, the twelfth after the Apostles, who then occupied the see.
In the second century we cannot look for much evidence. With the exception of Ignatius, Polycarp, and Clement of Alexandria, all the writers whose works we possess are apologists against either Jews or pagans. In works of such a character there was no reason to refer to such a matter as Peter's Roman episcopate. Irenaeus, however, supplies us with a cogent argument. In two passages (Against Heresies I.27.1 and III.4.3) he speaks of Hyginus as ninth Bishop of Rome, thus employing an enumeration which involves the inclusion of Peter as first bishop. Irenaeus we know visited Rome in 177.
In the first quarter of the third century (about 220) Tertullian (On Modesty 21) mentions Callistus's claim that Peter's power to forgive sins had descended in a special manner to him. Had the Roman Church been merely founded by Peter and not reckoned him as its first bishop, there could have been no ground for such a contention. Tertullian, like Firmilian, had every motive to deny the claim. Moreover, he had himself resided at Rome, and would have been well aware if the idea of a Roman episcopate of Peter had been, as is contended by its opponents, a novelty dating from the first years of the third century, supplanting the older tradition according to which Peter and Paul were co-founders, and Linus first bishop.
Source: The Pope (Catholic Encyclopedia).
Conclusion: It appears that the official use of the title of Pope (as given to Bishops of Rome) and confined to the successors of Peter, did not come into effect till the reign of Pope Gregory VII (1073-85).