If you are going to a business that cuts hair, such as a barber or a hair salon, then the person at the counter will likely say something like: "Hi, what can we do for you today?" You could respond with: "I would like a haircut, please." or "Just a haircut." (if that is all you wanted). Your three phrases above would probably be understood to mean that you wanted a haircut, but they don't sound natural. #2 could mean that you wanted to cut your own hair in certain circumstances since "I" is the subject, the one doing the action (cutting here).

All of the expressions work and number 2 could mean you do it yourself but it is more likely to be taken that you want your hair to be cut by someone else. To ask the hairdresser your request you would ask to have your hair thinned but keep the length.