Good morning, I would like to assist you today.
I find it very disturbing that you have taken your child to a child psychologist; and with her showing such signs of depression, they only say that she is immature for her age. As a parent, I definitely recommend to get a second opinion from another/ or other child psychologists; and specifically one that deals with depression with children her age. If you haven't already gotten the first psychologist as a referral from your pediatrician, I would recommend that you start there. A check up with your daughter's doctor can also be a way to ensure that there are no underlying physical health issues that may be causing your daughter to experience such depression. Here as well is a link where you can search by zip code for the kind of psychologist or therapist that is right for your family: http://www.goodtherapy.org/
I would not recommend punishing your daughter when she does make any form of comment about harming herself in some way; get her help immediately when this occurs. Outside of getting your daughter to the proper professional that will recognize her need for counseling, on your end, stay active with your daughter. Talk to her everyday, about life, and listen to her when she talks. Let her know that you are there to listen to her whenever she needs to talk; and always keep a calm and open ear when she does. If there are any activities that she enjoys, dance, music, art, computers, etc., get her involved in a program that will allow her to express her talents and gain confidence in herself and her own abilities. While dealing with depression in teens is a serious thing, it is also quite common; so know that you are not alone,and there is help available. Here is one more article that provides a littler more insight on how to deal with your depressed teenager: http://www.empoweringparents.com/Is-Your-Child-Depressed-6-Ways-to-Help-Them-Cope-Kids-and-Depression-Part-II.php
If there is any further help that you need on this issue, or if you feel that I have not provided you with a thorough answer, please message for earthsister. I wish you and your family the best.