If this is a serious debate, well so be it, I suppose the credentials
and typical type of jobs they do is most relevant. I mean, the 65 year old Italian guy that's been holding scissors in his barbershop for the last 40 years might be licensed to do dye jobs, but if I walked in and asked Luigi for a magenta streaked faux-hawk he'd probably kick me out.
Barbershops aren't just a place where you get your hair cut - some of them are, sure, but for a long time and in many places a barbershop was and still is
a social establishment. I'm not talking credentials and whether a girl can cut hair or any of that stuff, I'm just saying that the barbershop can be a place where just men go. The guys working there are men. You talk about local politics, sports, or you don't say anything. Some guys are just hanging out at the shop, you talk, you jive, you get trimmed up, and you come back next week to keep looking fresh.
In this sense, I can see what the ISTJ is saying. If that's your definition of a barbershop, then yeah, a man is a barber.
Also, barbershops have their own local cultural 'rules' and variations.
Some people are more set about having things a certain way than others. Also, maybe he's purposely giving you a hard time to make a point.