Welcome to the forums.
It can be really hard caring for someone with mental health issues. Especially when that person is a child of yours. I understand the need to want to fix the problem for him.
I think your son probably wanted you to read that notebook. I did that too when I was a teen. I didn't know how else to communicate how I was feeling and I didn't want to tell mum direct because I thought it would make her feel worse. So I would write it down somewhere and leave it out hoping she would read it.
People who want to hide their thoughts are much better at it. And if he truly didn't want help he would never have told you how he was feeling.
I do not know if it is appropriate for you to talk to his psychologist. They obviously won't be able to tell you anything about his sessions because of confidentiality. But if your son finds out you were talking to them behind his back that might make him feel untrusted and unable to be open with them.
You can always suggest that he contact BeyondBlue or Headspace or Lifeline if he is being troubled by suicidal thoughts inbetween appointments.
I hope the medication increase and the psychologist helps.
Kind thoughts Jess