I have read on this period of history and not just bits and pieces. The group in question at this time, which St. Stephan addressed was the acceptance of Novatians back into the Church. The Novatians were a Trinitarian group, who held the same beliefs about the Trinity as the orthodox Church did. Their deviation from orthodoxy had to do which matters concerning forgiveness and mercy and who deserved it.
Your believing the baptism issue was only over Novatian is to pick up a piece and ignore all the reset of the extant evidence about RC history.
Cyprian (source cited before):
2. He [Stephen] forbade one coming from
any heresy to be baptized in the Church; that is,
he judged the baptism of all heretics to be just and lawful. And although special heresies have special baptisms and different sins, he, holding communion with the baptism of all, gathered up the sins of all, heaped together into his own bosom.
Firmilian (next letter):
And indeed, as respects what Stephen has said, as though the apostles forbade those who come from heresy to be baptized, and delivered this also to be observed by their successors, you have replied most abundantly, that no one is so foolish as to believe that the apostles delivered this, when it is even well known that these heresies themselves, execrable and detestable as they are, arose subsequently; when even
Marcion the disciple of
Cerdo is found to have introduced his sacrilegious tradition against God long after the apostles, and after long lapse of time from them.
Apelles, also consenting to his blasphemy, added many other new and more important matters hostile to faith and truth. But also the time of
Valentinus and
Basilides is manifest, that they too, after the apostles, and after a long period, rebelled against the Church of God with their wicked lies.
No Polycarp didn't go there to correct her false teachings but rather to mediate between two parties.
Irenaeus was the mediator between Victor and Polycrates, but points out that Polycarp and Anicetus conflicted, yet lived in peace with each other.
"And when the blessed Polycarp was sojourning in Rome in the time of Anicetus, although a slight controversy had arisen among them as to certain other points, they were at once well inclined towards each other [with regard to the matter in hand], not willing that any quarrel should arise between them upon this head. "
ANF01. The Apostolic Fathers with Justin Martyr and Irenaeus - Christian Classics Ethereal Library
Irenaeus again:
He [Polycarp] it was who, coming to Rome in the time of Anicetus caused many to turn away from the aforesaid heretics to the Church of God, proclaiming that he had received this one and sole truth from the apostles,that, namely, which is handed down by the Church."
http://www.ccel.org/ccel/schaff/anf01.ix.iv.iv.html?highlight=polycarp,rome#fnf_ix.iv.iv-p6.1
IOW, Anicetus was unable to steer Rome away from the heretics. Anicetus was unable to proclaim the same faith handed down. It took Polycarp to do so. As mentioned above, however, unfortunately, rather than being grateful toward bishop Polycarp and learning the faith handed down from apostles, Anicetus still went his own way on certain matters.
Yes it does mean that Cyprian and Firmilian lost, and lost for a good reason. They were wrong.
You are saying that Marcion's or Arius' or LDS baptism is able to cause a person to be born-again from above, that they are equivalent to a Christian baptism. You are saying they posses the same faith, the same Father, the same Son, the same Spirit. The Catholic Church disagrees.
So, to the thread, one could say RCC was born in 256ad as it saw heretic baptism as equivalent to Christian baptism.
Or, one could say RCC was born in the period from 115 to 195 as it turned from the faith once handed down as the clash between Polycarp and others shows.