As you have probably learned, you want to establish a connection first. Given that you had suicidal thoughts yourself, you can start with something like "Yeah, man, some days I want to off myself, too. Life sucks. I swear, if I had a pill handy I might have done it already." You then try to go on by comparing the issues that make you think of ending it all (loss of a job, of a partner, bullying, depression, illness, ...), asking for advice in your circumstances etc.
There is no point trying to convince them of anything until they trust that you understand what they are going through. The lines you quoted ("You are not alone in this. I’m here for you." or "How can I best support you right now?") actually set you apart from them. What they hear is: "I am in a good place, I can even spare some of my happiness to listen to your clearly overblown issues".
Even after you are on the same wavelength, pushing what worked for you on them is probably counter-productive (typical-mind fallacy etc.). Letting them tell you what got them off the brink before and whether it would work this time might work better. Slipping into one of your more depressed moods and letting them give you advice is more likely to get them to apply this advice to themselves, a far more effective approach than rummaging through your own bag of tricks and see what works on them. There is, of course, a chance that their foul mood would push you over the edge, too, and you should be prepared for it.
There are helpful online resources available, such as the video testimonies from the It gets better project. Once you understand your friend's issues, you might offer visiting a relevant site together and talk it over while browsing.
If this seems like too large an investment of time and effort, it's because it is. Whether you find it worthwhile for you is up to you to decide.
As someone who has experienced depression, hearing that other people are unhappy doesn't help me at all or make me feel sympathetic. Having someone simply offer to help or care is what I am honestly hoping for when I'm feeling down.
This may not generalize well, but when I'm feeling down, I still have some sense that things could be better, and might be better again, and emphasizing that, and that people are around to pull me up from the bottom until they do, is valuable for me.
Last month, two people far at the periphery of my social circles have threatened suicide. Seems like a sign for me to learn some ledge-fu.
I reviewed the stuff I'd learned back in high school ("Listen." "Be supportive." "Don't argue." "Etc etc etc.") I have trouble believing that this would work outside of movieland, especially on strangers. More so, in person I'm an awkward, fidgeting introvert---the impact of everything I say is thus diminished, and I sound very insincere or clinical, like I'm following a bad movie script, when I say anything like, "You are not alone in this. I’m here for you." or "How can I best support you right now?" I doubt that this would sound any better in writing.
I suppose I could split my question into two related ones: what would you say to a person threatening to commit suicide, 1. in person, and 2. in an email?
I'm looking for out-of-the-box ideas that don't rely on charisma or compassion shining through. Personally, if I ever need to talk myself out of suicidal thoughts, I apply the "bum comparison principle": if my life is so crummy that I'm willing to commit suicide, then I should be willing to just walk out on everything I value and drift off in a random direction, survive by dine-and-dashing out of cheap restaurants and wash dishes if I get caught, maybe take odd jobs or hitchhike or gather roots and berries or blog from public libraries. I don't see this possibility in a negative light, and yet I still haven't done it. To me, it means that however bad my life may seem, I'm still too attached to it to walk out; therefore, suicide isn't on the menu.
People have different reasons to want suicide, and I understand that what works for me with my first world problems probably won't work for a person who is in too much physical pain from an incurable disease. To the best of my knowledge, the two people I mentioned earlier are both unskilled laborers who had lost their jobs, one of them so long ago that he's no longer eligible for unemployment benefits. I don't think I'll meet these particular people again, but I'd appreciate everyone's thoughts on what I could've said if my brain hadn't frozen.