A BARBER cuts men's hair. A HAIRDRESSER and A HAIRSTYLIST cut both men's hair and women's hair. Is that correct? And are all of these words commonly used?

Barber is primarily for men’s hair, and to that extent I’d say anecdotally I associate hairdressers with women’s hair. Hairstylist can be either or.

All of those words are commonly used. Nowadays, a barber isn't strictly only for men, but a barber usually does specialize in men's hair

I’d say that this is generally correct, but especially “barber” and “hairdresser” are complex cultural words that may not translate cleanly into other languages. Not only do barbers usually only cut men’s hair, but they are usually men themselves and offer other services like shaving.

'Barber' is an old word, and it has an old-fashioned or traditional connotation. It suggests a man who cuts men's hair in short, traditional styles, and maybe also shaves men's faces, probably with a straight razor. I don't recall ever encountering a woman who described herself as a barber. Barbers used to have some overlap with surgeons, and would perform bloodletting. (I think the traditional red-and-white barber's pole is supposed to suggest blood, but I'm not certain.) Barbers have not performed bloodletting for a long time, but, as I say, it's an old word. Hairdresser and hairstylist are newer words without the cultural weight 'barber' has. To my eye, they are people of no particular gender who cut and style hair belonging to people of no particular gender. All of those words are commonly used.

Hairdresser and hairstylist are for women's hair. Barber is unisex.

A barber is a hairdresser who specializes in men's haircuts and also does beards and is therefore usually associated with men. But I do know some women who go to barbers also. Hair dresser and hair stylist are more associated with women due to stereotypical gender norms. But again I know plenty of men who go to one. I am a man with very long curly hair. I go to a hairdresser. All that said, barbers are less common overall in my experience. They're common in certain communities, but not in general. At least in the US. But to answer your question, yes, all 3 words are used commonly.

“Barber” derives from Latin “barba” meaning beard, so a barber is primarily for men’s hair and facial hair. Hairdressers and hairstylists primary focus is hair atop the head, so they can do anyone’s hair so long as they’re familiar with their hair type and the style the client is looking for.

Generaly barbers do short hair and hairdressers do long hair.

A barber might cut a woman’s hair if she wanted a traditionally masculine haircut

Yes. Plus cosmetologists who cut hair, do nails, eyebrow waxing, etc for any gender. 

On the high street yes, you're right, but in more specialist areas - I work in film/TV - a barber is specifically a hair cutter, not just a brush & hairspray wielder. \[I'm not trying to demean the stylist, just make a more obvious distinction\]. It's not divided by gender, but by profession and training. A hairdresser's high street store can have both, but you may not actually really see the distinction unless like a Tony & Guy type place they will have 'stylist' & a 'senior stylist'. They will also have colouring specialists.

I’m in central Canada. Barber is predominantly the term men’s haircuts in the usage with which I am familiar. It could also be women’s hair if it is only a cut, but I wouldn’t expect a barber to dye hair, put a permanent in, or other more complex hairstyling. They are also expected to be the place to get a beard trimmed. Hairstylist seems to be now be the dominant other term. When I was a child (25-35 years ago), I often heard hairdresser but I don’t hear that term anymore. I have only heard haircutter in UK media, not locally. I expect a hairstylist to do all kinds of other hair treatments and to usually start with washing your hair. They do perms, dying, highlights, extensions, and so on. Often the hairstylist also has an aesthetician who works from the space doing nails and so on. My local barbershop doesn’t even have the ability to wash out your hair.

The word barber is derived from the Latin word for beard, so you can see why it is primarily associated with men. The word literally means "bearder" (one who removes the beard). That said, historically barbers had other tasks for instance in the middle ages many of them would perform medical procedures like bloodletting and even surgery. That said, a good shave has always been the focus of the barber, though they're usually all hairstylists today as well. A woman can go to a barber, but they may or may not be skilled with feminine hairstyles.

Barbers primarily work with very short hairstyles and beards, using buzzers and razors. Hairdressers and hairstylists usually work on longer hairstyles. The type of work done is quite different. There is some overlap, but usually barbers tend to work with cuts that are short enough/use product to stand on their own and resembles "trimming the hedges" more than hairdressers. Hence why black men with short hair tend to almost exclusively go to barbers, as barbers are more in their element with tight, curly, short hair.

In many states of the US, they’re regulated professions, making the distinctions more important. In my state, barbers are trained to cut hair and to shave people, and to perform some chemical treatments of hair. That set of skills makes them typically specialize in men’s hair. Hairdressers/stylists typically have a different license (in my state, cosmetology) which might give them expanded abilities, but does not allow them to shave people.

Barbers also cut hair dry. Hairdressers and stylist may wash your hair before they cut and style it. Hairstyling includes things like permanents and hair coloring.

Barbers were known for shaving and pulling teeth as well.

I would also mention that, in the US, there's a legal distinction. There are two separate licenses for cutting hair professionally in the US: a cosmetology license, and a barbers' license. The training and exams for a cosmetology license also includes hair dying, nail treatments, and makeup; the training for a barbers' license also includes a focus on straight-razor shaving. People who call themselves "barbers" must have a barbers' license, but someone with either license could be a hairdresser/stylist. Not usually important, but it means anyone cutting hair can choose to call themselves a hairdresser/stylist, but not everyone can call themselves a barber. I'm not sure if other countries have analogous legal distinctions.

A female barber is called a Barbara. I'm just kidding that's not true

I am a man. I have never said hairdresser or hairstylist. Well I don’t know if I’ve never said it but I’ve never said off to the hairdresser I don’t think I say Barber either. I have to go to The Barber? It’s usually where you’re going? I’m going to get a haircut. Where are you going? Barbershop. Supercuts. Sports clips. If I ran into the guy that cuts my hair at the store or something, and my wife said who is that, I would say that guy cuts my hair. I wouldn’t say that’s my barber. That having been said, all of the words and examples that you gave work. A hairdresser or hairstylist can cut a man’s hair. I assume that’s what they’re called it like Supercuts or sports clips. They aren’t barbers. As a matter fact, if I go into a barbershop and ask for a shave, they will tell me no if The Barber isn’t there. As if The Barber is specifically for shaving. I know that it’s not. But sometimes it is? As a man I would never say I’m going to the hairdresser or the hair stylist. Mostly because they aren’t doing my hair. It’s not like a color and a blow and a cut and whatever else a hairdresser does. It’s just a haircut. Barber cuts a man’s hair Hairdresser cuts both but the word is used more for women. But if someone works at a salon where they cut both men’s hair and women’s hair, they would say I’m a hairdresser not a barber. To me a hair stylist does something different. It could involve a cut. But you’re styling the hair. Like putting your hair up so that you look like you’re in the 40s. Men can have that. Actors definitely have that. But I have never paid someone to style my hair. Those are just my initial thoughts. I’m not a linguist

Barbers do generally cut men’s hair, but they’ll also sometimes cute women’s hair if they have more masculine hairstyles. Hairdressers and hairstylists cut everyone’s hair, yes, and generally have more training on different hair styles and other aspects of hair styling (such as hair dyes and extensions, etc…). All three words are used, though I hear hairstylist or even haircutter most I think (UK) and barber has connotations where I live because it’s associated with Turkish barbers- long story on that one

There's a YouTube channel called Psychology in Seattle, and it always makes me chuckle how he says "hair cutter" for someone who cuts his hair lol. Yes, barber is for boys, and the other ones you mention could be used for either imo.