He was pointing out that some were correctly saying they were of Christ, or in Christ. The problem was the ones that were saying they were of Paul, or of Apollos, or of Cephas.
***The problem was the ones that were saying they were of Paul, or of Apollos, or of Cephas.***
But you stated, "The faction was caused by the identifying with the one who baptized them."
And though that is a possible interpretaion, it can not be proven from the text - because the text does not state in which manner they were identifying with the varous names.
They may have been identifying as disciples - as in "I am a disciple of Apollos" or "I am a follower of Cephas" etc.
You infer baptismal identification from the text.
This interpretation is made even more unlikely because of Paul's additional reference to "crucifixion".