Can pychiatrist have you commited if you tell them you cut yourself?

Typically no, unless it was an attempt to end your life. Usually the doctor will ask you if it was a suicide attempt and go from there

No, if you're under 18 years of age, then you typically aren't committed for self-harm. It's when you either threaten suicide or if the self-harm looks more severe then you'll more than likely have to go to the emergency room for an evaluation. Sometimes, they'll ask if you can stay safe for the next appointment (which can be months out) or if they can call later that day, or email and check in on you. Other times, they can contact your therapist and ask that they check in on you and see how you're doing. It depends on the relationships between everyone.