I think it's important to mention this:

There are not simply two kinds of suicidal threats (i.e. serious vs call-for-help).  There's a lot of gray area, and a lot of unpredictability.

Take the person I met that drank a cup of bleach.  They will tell you all about it.

If there's one thing I've learned from working in a hospital, it's that every person who threatens suicide is in a world of problems.  You take absolutely every threat seriously because "serious" is not simply limited to the actual act of attempting suicide.