I know that most people here will discourage you from sharing, and it makes sense why they would. SS feels like a space to protect and letting mental health professionals know about it feels counterproductive. Further, it could put you at risk for getting committed or having your freedom restricted to some capacity. But I think you are posting this here because you really do feel like it's the right thing for you to do. So, it may be the unpopular opinion, but I say share.
I shared with my therapist. He is extremely receptive and understanding of these things. I've gone as far as telling him extreme detail about my plans and methods and he never once had me committed. His help actually ended up getting me out of that really dark, urgent place. It is a risk that I took, and it paid off. This will be a risk that you will take, too. If it feels right within your gut then I think you should.
Personally, I told my therapist in a very slow build up. He knew for a while that I was struggling with thoughts of suicide without really knowing if I was going to execute it. My childhood best friend actually found this site first, and next week is her two year death anniversary. I started off telling him about how she found the forum, and how it broke my heart that she passed away. As time went on, I began hinting that I was getting more serious with my thoughts. That's when I told him I browsed this website in a moment of temptation and that I was really getting close. Now, he knows that I am an active member here (he does NOT know the name of the website), and is actively encouraging me to get off of it. I'm not really talking about it much now because I know it isn't something I want to be talked out of at the moment.
In terms of what you want to say specifically, that is completely up to you. Whatever feels right. I elected not to give my therapist names or identifying information about SS, but I did tell him it was a forum that was pro-choice on the subject of suicide, and offered resources and support for those who want to end their life. I also said that that was the means in which my best friend got her materials for her death, and how everyone on the forum is open minded to death being the best option for someone. It was very relieving to not live this in complete secrecy, but also scary to open it up with someone. Overall though, I know I have the freedom to talk about it when something comes up. Like recently, I thought a member here passed away that I was on the phone with, but turns out he survived. It was a lot of whiplash and really scary for me but I have someone I can talk to about it that knows.
All in all, you will know what feels right and what you want to keep to yourself. You will navigate through it once you start talking. It is not your responsibility to protect SS, and consequences that may come from sharing are all a part of the risks you take. You will know if it is worth it when you get there. Please reach out if you need someone to talk to.