How can I make the therapist I'm about to see, think that I don't need help?

You will only go to a psychiatric hospital if they feel you are a danger to yourself or others. Or if you meet other criteria's.. In your case if you are cutting to sooth you shouldn't be hospitalized, but you should start looking for other coping skills which a therapist would be able to help you with. So be open, kids are usually only hospitalized on the spot if they need medical stabilization.