You're right about "barber," Doomer, but I disagree about "hairdresser."
I would say that a barber has male clients; a hairdresser has female clients; and a hairstylist accepts both men and women.
Or to put it differently: A man never says "I'm going to the hairdresser", and a woman never says "I'm going to the barber". But both sexes can say, "I'm going to the hairstylist."
Also, a barber shop takes only male customers, and a beauty parlor takes only females, but a hair salon or styling salon usually takes both males and females, although some salons mainly advertise to women. And in the US, traditional male-only barbershops and female-only beauty parlors are not as common as they once were -- the trend is towards "unisex" salons where both men and women can get прически, маникюры, и педикюры.