I already found the clip but I don't know when it aired. The story of how I discovered this segment actually comes from a personal home movie and not from a VHS recording. I used to have episodes recorded from PBS but I don't anymore. Now for a rather long story. My parents used to film me at various stages of my life. You know for posterity and sometimes I like to look back at them. I guess I was about 2 years old and this was filmed in 1994, so I imagine it must've been a rerun of Sesame Street I saw.
At first I was listening to Michael Jackson's Thriller on a Cassette tape which I used to do a lot back then. Suddenly my mom asks me if I want to see if Barney is on. Believe it or not I knew how to work the remotes in my house at that age, so I turned on the TV in the middle of the Fixin My Hair segment. I had no recollection of seeing this until I saw it on the original tape used for filming. My dad used to take some of these tapes that were meant to be played in an adapter (which I still have) and put them on a VHS, so I guess some stuff got edited out.
Upon turning on the television and seeing this segment, my mom says to me "You missed Barney. This is Sesame Street.", then a split second later Bert comes on after "Fixin My Hair" ends. I have no idea what the Bert segment was so I can't really describe it. I guess I decided I wanted to listen to Thriller some more or something because my mom on that tape can be heard saying, "Oh! You turned off Bert!" lol. If anyone remembers "Fixin My Hair" and possibly the Bert segment that came after in this particular episode, I would love to know.
I also have another home movie related memory of Sesame Street. I was either 4 or 5 in 1996 and I caught a bit of a sketch where Cookie Monster was arguing with somebody over who gets the cookie, cause I guess there was only one. That's all I really remember.