It sounds to me like further clarification of the contract with your therapist is necessary. If your SI thoughts don't result in SI urges, in other words, if they don't transform into SI feelings, it may not be necessary for you to contact her immediately as they come.

It's understandable when therapists get a little anxious around client's suicidality. Just like anyone else, they feel the natural urge and also the professional obligation to prevent client's from hurting themselves. But sometimes this urge, as understandable as it is, may obstruct the natural flow of therapy because it gives the client a message that their SI is something abnormal and to be scared of. It is necessary to attend to suicidal urges immediately, but as long as the suicidal contemplation is in the realm of thought only, it can be processed in therapy just like everything else.

SI, in and of itself, is not something abnormal and something to be scared of. It's a natural part of life and it's a very common occurrence. We all have those type of thoughts cross our mind when things get tough and we start wondering "what's the point of all this?"..