The suicide prevention hotline is never your friend.
I've called the cops once, on a woman who had taken two bottles of NyQuil and was losing consciousness with the phone on speaker begging me to stay on the line until she died because she didn't want to die alone.
I've talked a dozen people out of nooses, helped teenagers get through the night with out cutting themselves, helped women who were being beaten by the spouse find a women's shelter. I've talked to the same depressed and lonely old women in a California nursing home almost every week. I spent two hours on the phone with a man who was driving because he kept threatening to crash into on coming traffic.
From what I can assume, your difference is on whether they’d harm others. I’ll assume your creed is libertarian, though there are multiple essays where respected libertarian philosophers have argued that libertarianism doesn’t allow you to give up your liberties (e.g. suicide and indentured servitude)
The fact that these people could be talked out of their suicides (and in fact that some wanted someone to talk to lest they hurt themselves) shows they clearly were having a temporary lapse in judgement. If we accept utilitarianism as the basis of libertarianism, living another day with a clearer state of mind increases utility.
The last one is bad because those people unfortunately chose to live that day.
People should have the right to die, but you're basically saying that people shouldn't be allowed the option of talking things over with a sympathetic listener before making a final decision.
Edit: I have attempted suicide before and self harmed many times. I dated a woman who attempted 12 times.
It turns out life can get much much better. I haven't self harmed in years, she's very happy and hasn't either.
It's very hard to know what living well feels like until you've done it. It took years but I like waking up in the morning.