I don't know the answer to your last question, and it's an important question. If I was in your situation, I'd ask my T about that. In a general "What can/will you tell my parents without my permission?" kind of way. You need to know that anyway, I think. How can you talk about ANYTHING if you don't know where the information might go?
Beyond that..... I had exactly the conversation you're talking about with my T a year ago. Well, I'd never called him, told him I was about to kill myself, and got hospitalized. If I'd done that, I'm sure he'd have done the same thing, but I don't think that would have affected the conversation you're talking about much. Except that we probably would have had it THEN.
Anyway, I told him the truth and he believed me. Now, part of that conversation was him, telling me he wanted me to call him if I ever WAS planning on killing myself. I told him I wouldn't promise that, because I wouldn't lie to him. I told him if I ever wanted to kill myself, then he'd be one of the LAST people I'd call. Sorry. He wasn't happy about that, but he wanted the truth.... Anyway, I told him I didn't, at that moment, have a plan, but that those thoughts could be pretty relentless and I got sick of listening to them all the time. That that was actually the main thing that led me to contact him to begin with. (Yep, I was seeing him for several months before I actually told him the specific reason I decided to check out therapy.) He said that there IS a difference between thinking about suicide and actually being suicidal. But that both were to be taken seriously.
Beyond that, on the subject of suicidal ideation, he said that he thinks everything thing our brains come up with is, or was, adaptive in some way, at some time. It might not have been a great idea, even when it got started, but it had a purpose. He suggests it's best to learn, and to operate with the best, most up to date and accurate information we can. So, he recommended I look at the possible origins of that suicidal ideation, to see if maybe I could see where it started and how it might have been adaptive then. Then to see if maybe I needed to up date some information.
I thought he was nuts. Truly! But, mostly to humor him, I tried. And, I came up with something. Something weird and unexpected, but I think it was the origin of it , in my case. Now that I know that, and have given some thought to where the ideas came from. How they did, in a weird way, make sense at the time, but don't actually make sense, the thoughts don't come up nearly as often.
I'm sure there are multiple ways to approach this, so it's best you talk to your T. I'm guessing she gets that there's a difference between thinking about suicide and planning it.