Sorry but I don't believe either is true.All Bible writers were Jewish, btw.
But the first schism in the second century CE was between antisemitic western churches under the leadership of Victor, while Polycarp, a quartodeciman, and the second century eastern churches followed apostolic teachings.
The Bible writers were highly educated, Greek speakers who took their Judaism from the Septuagint. It seems more likely that they were converted gentiles since their understanding of Judaism is limited. Also, conversions from Judaism were always small in number.
The first schism occured in the decades after Jesus' death and concerned the question of if Christians had to first become Jews. Paul said no while James said yes. The matter was likely decided by the Romans when they sacked Jerusalem. If James' faction had won, Christianity would still be a minor cult within the minor religion of Judaism.