As a thought experiment, I've tried to put myself in someone else's shoes (like a counselor) and wonder: what would I say, what would I do if someone like me came in and said they wanted to off themselves? Or what would I do if it was someone I already knew, a family member or friend?
And well, I don't really know. I'm not sure there is a right way... as soon as one starts talking about "right ways" we begin trying to put all suicidality into a neat and convenient paint-by-numbers box, when it really needs to be a case by case basis no matter what we think we might know in advance.
I also don't fully understand what kind of training and education the average counselor undergoes, and how much if any of it pertains to handling cases involving suicidality. I also assume counselors or similar professions have certain obligations that, if they don't meet, can potentially make them liable if something happens, or at least leave them open to some kind of punishment. The first thing any human will do is practice CYA. For example, in some places people are afraid to intervene if somebody is having a life threatening health emergency in public, because the situation can go wrong and they'll potentially find themselves in serious trouble in the future despite their obvious good intentions.
The best I can come up with, is that I'd need to do my best to try and understand the other person first, to listen to them, to withhold judgment and knee-jerk reactions, to try and get some idea of where they're coming from and why they feel the way they do. The basic logistical difficulty is that this takes a lot of time, and in some cases it's hard to say how much time you have to try and work something out. This involves asking questions, getting the full story, the full picture, and the challenge of trying to level with someone who is in such a dark place, neither encouraging nor invalidating them. That's part of why I don't talk about these thoughts or* feelings with other people; the vast majority are *not* equipped to handle it. Our "support" systems are actually very ill-equipped to handle these issues, in my experience.
The "help" you'll likely receive: take this pill, go away, come back next week; shortcuts like guilt-tripping (what about your family, and other whataboutisms); or platitudes tantamount to "you just need to be more positive." There's also the issue of what can they even do? What do I expect some stranger handicapped by a code-of-conduct of their profession to be able to do for me? Give me a million dollars? Give me a job that doesn't make me want to die? Hook me up with some hot single person in their family? Like... I come to the conclusion again and again that there isn't anything practical, anything tangible they can even do. So if I need something more than paying someone to listen to me vent, I might as well not bother.
I'd seen a dozen therapists before I became too old and lost my parents' insurance. Only two I lasted with for a year or more. It's really, really difficult to find someone who you feel like you can trust, and it often takes considerable time to build that client-counselor relationship.
Perhaps the first thing to do, is to do nothing. And I mean that more figuratively. To resist the gut instinct that says "this is dangerous, I must do something right this instant, I must say something that will shut this down," like they're defusing a ticking time bomb in a shitty action movie. Again, perhaps the first thing to do is "nothing."