I think something a lot of people don't understand is that depression is not suicidality, and suicidality is not depression.
People can, and are, depressed without being suicidal, and sometimes suicidality peaks as people are emerging from depression.
Suicidality is a wave, and the trick is to allow that wave to crest and subside WITHOUT acting on it.
Whatever it takes to ride it out. For some people that's distraction, like watching television. For others it's calling a friend -- not to talk about the suicidality, but just to talk. For others it could be as simple as going to sit in a coffee shop or library, because the presence of other people is a huge diminisher of suicide risk.
That's what suicide safety planning is about. It's like having any other type of emergency plan, like a plan for fire or evacuation. It's making a plan when you are in the frame of mind to do so, so that you can just DO the plan without having to think about it when the occasion arises.
When you're in the midst of suicidal ideation, or even intent, you're not in a problem-solving mood. So knowing past!you, with the help of a therapist hopefully, came up with the plan and all you have to do is follow up until the wave crests and subsides, is what allows you to see another day.