If you look at the early issues of WIRED magazine, I would say that's the post-ANSI aesthetic, but that's just my opinion. 1994 and the Internet really killed off the BBS thing fast, otherwise I think Ripterm (or something like it) would have replaced ANSI because modems were getting faster http://www.bbsdocumentary.com/library/PROGRAMS/GRAPHICS/RIPS... For me, 1994 was switching from primarily Pascal, TheDraw & DeluxePaint to HTML, Photoshop & Java.
As far as the demo scene, I was not really part of that but I think (correct me if I'm wrong) that was based more on the C64/Amiga which had powerful graphics/music capabilities way before the PC. So I would guess the demo scene came first and the ANSI "BBS" art was more a product of the IBM PC-compatible world. I remember going to an Amiga club meeting and a few of these guys had big boxes of disks and they would just trade like that, they had limited Internet access through the local university but they didn't have a BBS.
I offered to host their MODs (Amiga music modules) on my BBS. Because of that, I had the largest collection of MOD music in the area. It was like Napster before Napster ;-)