Was Peter's Son John Mark who Wrote the Gospel of Mark?

[Nottheworld]

"She who is at Babylon, who is likewise chosen, sends you greetings; and so does my son Mark" (1 Peter. 5.13).
"She who is at Babylon" refers to the saints in Rome.

My study notes read as follows:

Mark, also called John Mark, was known to many of this letter's readers because he had traveled widely (Acts 12.25-13.13; 15.36-41) and was recognized as a leader in the church (Colossians 4.10; Philemon 1.24). Mark was probably with the disciples at the time of Jesus' arrest (Mark 14.51,52). Tradition holds that Peter was Mark's main source of information when Mark wrote his Gospel. Barnabas was Mark's cousin.

[JakobG]

Though Peter was married, it is doubtful that Mark (Marcus) was Peter’s "biological" son. We learn in Acts 12:12 that Mark’s mother was Mary and that her house was in Jerusalem. (Peter was from Galilee.) Also, Paul describes Mark as Barnabas’ cousin, not as Peter’s son. The word “son,” has often suggested affection and a spiritual relationship. Mark may have become a Christian under Peter’s ministry.

[Nottheworld]

I agree Peter's "son" Mark is a spiritual son like when Jesus at the cross said to His mother, "Woman, he is your son" (Matt. 19.27) referring to John, the disciple whom Jesus loved.