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The opinion that this Mark is a different Mark is found in the writings of Hippolytus of Rome who thought them to be separate people. According to Hippolytus, in his work On the Seventy Apostles, Mark the Cousin of Barnabas (Colonel 4:10. Phlm 24) is distinct from John Mark (Acts 12:12, 25.

13:5, 13.

15: 37) and Mark the Evangelist (2 Tim 4:11). They all belonged to the Seventy Apostles of Christ (ranked #56, #65, and #14, respectively), who were sent out by Jesus to saturate Judea with the gospel not long before his crucifixion (Luke 10:1ff). (There are three possible sites for this place: one in Greece, one in Thrace, and one in Cyrenaica)

Paul mentioned Mark the Cousin of Barnabas explicitly in Colossians 4:10 (Bible Quotation from New American Standard Bible 1995 Update ):

23 Epaphras, my fellow prisoner in Christ Jesus, greets you, 24 as do Mark, Aristarchus, Demas, Luke, my fellow workers.

(Phm 1:23-24 NAU).