Thank you for asking that question. I’m sure there are many other people who share your concern but didn’t write to us.
Therapists regularly talk with their clients about suicidal thinking. If someone reports a story like yours to me, it doesn’t automatically activate a referral to hospital. What it does activate is a series of questions to help us both determine if the person is safe. Often the incident was a serious way to show both of us how deeply troubled the person has been. Most of the time, the person didn’t want to die but wanted (and still wants) the emotional pain to stop. Then we talk about that. Such sessions often deepen the trust between us and take the therapy to another level.
If, on the other hand, we agree that the client isn’t safe, meaning that the client may make a more drastic attempt, then we will discuss whether and how a hospitalization would be helpful. (And, hard as it may be to believe, I’ve had more than one client write papers for school while hospitalized. They found that feeling safe in the hospital let them focus on doing the work.)
If you are worried about how your therapist will react, you could ask her something like, “What do you usually do when someone talks about suicidal thoughts or even maybe thinks about doing it?” the answer may reassure you. It is likely that you will open a new and useful path in your treatment.
I wish you well.