A psychologist is going to look at whether or not you meet the criteria for a diagnosis of Gender Identity Disorder (GID). If you meet the criteria, then things proceed. If you don't meet the criteria, he might start delving into other areas. Although it's not at all necessary, I'd suggest your being familiar with the criteria, and ask yourself, what can I honestly tell him that will support this diagnosis? You should be very honest, and not attempt to hide anything.
I imagine he would want to know things like, How old were you when you first identified as female? What does dysphoria mean to you? What did you do, or have you done, in an attempt to try to reduce the symptoms of dysphoria? What do you think makes you different from other males? Or, what makes you think you are more like a female? The truth is, a good psychologist is going to have more and better questions than I could even begin to imagine.
Your statement that "He mainly focuses on the individual's mental ability to handle a transition and also the maturity and certainty to go through with it" makes me wonder why your parents are taking you to him. If they don't trust the diagnosis from the other psychologist, then it would seem to me that they certainly want much more than to know your "ability to handle a transition and also the maturity and certainty to go through with it." It seems to me that they want first to know if you really do have GID.
Also, you're asking for advice on what to tell the psychologist to prove that you are mature enough to handle the transition. Well, who here really knows how mature you are? The thing is, though, that transition is not about maturity, it's about treating a condition for which there is no other treatment (this is looking at it strictly from one aspect, only), and, except for where treatment is first required for a co-existing condition, I don't know of any accounts where later transition was better than earlier transition. I think maturity takes a back seat to treatment.
Assuming you are indeed suffering from GID dysphoria, I think you really don't have anything to worry about.