In BrE, the hair that covers your forehead (or part of it) down towards your eyes is your fringe. I had never heard of "bangs" until an American friend of mine said it when I lived in Spain. I wondered what on earth she was talking about! So if you're getting your hair cut in the UK, you'll want to ask them to cut your fringe fairly short. From what I can see, you want "An inch off the top, number two on the sides and keep my fringe fairly short". And no, a "barber" in the UK does not imply a cheap haircut. It's just the term for a men's hairdresser.
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