@Dreamy 

From what I think and what I've learnt so far, no, I don't think rhey shouldn't push you to go into details, or anything you're not comfortable going into.  They may ask specifics on what someone has been thinking, or what they've olanned in regards to harming themselves to see what sort of risk the persin is at.  But going into the sort of details you mentioned seems completely unnecessary to both offing support and determining the level of risk you may be at.  All I could see it doing is upsetting anyone, pushing for details like that, making them feel uncomfortable and possibly unsafe and would probably make them very hesitant to reach out again.

That's just how I see it anyway.  I don't see any way it would help to push for details like that.