Where on Columbia's site can that be found? Because in the Prospective Students section (from which you start to get to any of the applications) it lists CC, SEAS, and GS as the only "Undergraduate Schools" at Columbia, with Barnard listed as an "Affiliated School":
Columbia University: Prospective Students
And as for how a Barnard student should best introduce herself...isn't it best to tell the truth before talking about any affiliation with Columbia? Say "I go to Barnard"; if the person you're speaking with doesn't know what that is, you can say simply "It's the women's college affiliated with Columbia University" or even "the women's college
of Columbia University". That's easy enough, no? Somehow I don't buy the argument that Barnard girls
need to start out by saying "I go to Columbia" just because not everyone knows what or where Barnard is.