Barbers and Hairdressers are Very Different
I'm surprised this hasn't been mentioned, but in the US at least, barbers and hairdressers have very different qualifications and licenses for historical reasons. Visible differences are barbers use sharp implements without guards, e.g, they shave people with straight razors, and use clippers freehand where hairdressers use guards of various sizes, and of course hairdressers do much more than just cut hair.
Also, barbers display a barber pole in front of their shop.
I'd like to see shop=barber