I’m sorry that your son is going through such a difficult time — which also means the family is, too. Without knowing your son, I’m not sure that I can help decipher things any more than you already have. The problems could be typical teenage angst and moodiness that can be influenced by hormones, developmental changes and social pressures, but it could also be a legitimate mental health issue. And you are correct in thinking that some teens talk about suicide as an attention seeking behavior, but it can also be a true expression of suffering. The combination of teenage impulsivity and depression can create a very serious risk.
It sounds like you have some resources in place with the therapeutic school and psychiatrist. Your best bet is to reach out to them and become as involved in his treatment as possible and make sure that he is getting the level of care that he needs. In addition, I would suggest helping your son find outlets that he finds enjoyable, build self-esteem and give him something to look forward to. This could include sports or clubs, volunteer work, martial arts, or it could be something like equine therapy or Outward Bound.
I hope this helps and things get better soon.
All the best,