Speaking from personal experience, if your t calls the police, your name goes out over the radio, and people who like to listen to the police scanner will know that your t thinks you are suicidal and has sent the police to check. In my case, that resulted in several people that I know of now knowing that t thought I was suicidal, and there are probably others. I say probably others because they may have heard it over the scanner, or may have heard it from people who do.

I think I would rather just my boss know. And t1 would definitely call the police again if I was a no show. He'd probably try my H first.