Rather than to engage in a back and forth, let’s just take the horse’s mouth:
https://barnard.edu/facts-stats
“Barnard is considered
one of Columbia University’s four undergraduate colleges
and students have access to libraries, facilities and activities on both campuses; Barnard students compete in NCAA Division 1 athletics through the Columbia-Barnard Athletic Consortium; Barnard and Columbia undergraduate students cross-register in courses at both institutions; and students receive a
diploma from both Barnard College and Columbia University.”
Then scroll to the very bottom of the page to the Copyright line.
PS: You are correct as far as governance: Barnard is not controlled by Columbia U’s board, but retains its indepedent Board of Trustees.