There are many statements that patients make that refer to suicidal ideation, which are not a threat of suicide. Such as, I dont want to wake up, I dont want to do this anymore, I wish I were dead, etc.
The therapist or Dr would generally assess the situation, such as, do you have a plan and the means, such as pills, firearms, a plan to jump off a bridge, etc. There are many questions they can ask to assess the lethality. There are identifiable risk factors. Most people that attempt suicide do not even want to die, they want to escape the pain and are out of resources and coping skills.
Often if you are having these thoughts, the T will help you with a safety plan, and that if it gets too bad, you will call them, go to ER, etc. Some therapists do a contract with a patient. Most do not want you hospitalized unnecessarily. Especially since they can only hold you for 72 hrs for evaluation and if they do not observe risk, they must let you go.
I am sorry you are feeling so bad. If you do have a plan to end it, it is best to share it with your therapist. I dont know your reason for not wanting the meds but I know that after 6 months I just went back on them last week. I need them-I was feeling suicidal. I feel much better already-not up to par, but not a risk. It is very hard living alone and having no supports when things get really bad, and being unable to get out of bed and nobody to advocate for you. Often the T will bring in a family member or close friend who has daily contact with you so that they will know what to do if things get worse for you.