If you say you have a plan, they ask if you have intention to follow through on the plan. If yes, then it's imminent, and so they consider that they have reason to have you psychiatrically incarcerated to keep you from doing harm to yourself.
A way around all this is, if you're suicidal, to work on the root issues that cause it, not the symptom of suicidality. Not to minimize the seriousness, but in a way it can be a red herring. If you have plans, then they want to prohibit you from acting on the symptom of suicidality, but the cause is not addressed and continues to have negative impacts in a variety of ways, set off other alarms. If you don't know the root cause, you could tell them to set aside the false alarm of suicidality and help you find the freaking cause, not try to shut up suicidality with CBT, but use methods like EMDR, trauma based protocols, etc. to get the source of the problem. If they're not willing to or capable of doing this and they just want to put out fires, then I think a more capable and less reactive therapist is in order. I remember reading about a therapist who worked in in-patient and a client was raging and threatening to throw a chair at her; she sat down and said she wasn't scared, and would the person like to talk about what was actually going on. The patient chilled out and felt heard because the therapist didn't take the bait of the threatening alarm. Heck, maybe we could do something like this with ourselves if we feel like suicidal thoughts are in control?