Both the PCP and specialists can sign paperwork for FMLA/CFRA. There is no separate mental health leave under FMLA. The provider does not have to disclose your diagnosis to the employer. In fact, the provider is specifically warned not to disclose the dx without your express consent. HR does not need to know that the leave is for mental health reasons. It is your choice whether to tell the HR.

The person who should sign the paperwork is the person who is treating you for the condition(s) necessitating the leave. If it is your PCP, then the PCP, but if it is the psychiatrist, then your PCP will likely defer to the psychiatrist and refuse to sign the paperwork because the PCP is not intimately familiar with the details of your case and cannot state what the prognosis is, for how long you need the leave, etc.

Also, the provider who signs the paperwork needs to state when you last saw said provider for the condition necessitating the leave. This is another reason you should only ask the person actively treating you right now.