This may depend on the country but I as a therapist ask everyone anyway. And I’ve experienced many, many people over the years being afraid of speaking up. It’s always a moment of relief when it’s out there and they realize I’m not freaking out over it.
I’ve pretty much heard it all. Including the various ways people try to approach the subject while still unsure how I will react. And I do think that is something you could try if you’re unsure about your therapist - talk to them about your suicidal thoughts and see how they react before you confirm plans or attempts.
Chances are of course they can get quite a bit from your way of talking about it, because you’re definitely not the first person with those thoughts in front of them. The thing is - suicidal ideation is, depending on the type of disorder, quite common. If we’d admitted anyone who thought about suicide to a psych ward immediately they would be bursting at the seams and we’d get nothing done at all. So that’s not happening. As long as you can convincingly agree with your therapist on a plan forward (which could mean: Okay, I promise not to kill myself until next Tuesday) you don’t have to be admitted if you don’t want to. Which also would be an option of course. Psychiatric wards are emergency departments. They are supposed to be there for you when you’re seeing no light at all and in my experience, at least where I live and work, in fact have saved quite a few lifes.