Totally agree. You are not being too sensitive, she is trying to wear you down. You said you've talked about these things multiple times, but she keeps bringing them up. Use your opportunity for quick notice (24 hrs) and be done. Just imagine, if she's acting like this now, while on a trial period, imagine what it will be like once the trial period is over.

Dear parent,
This notice is to inform you that as of (date) childcare for (child) will no longer be available. As per the contract signed on (date), please consider this is your 24 hours notice.
Dawn2Dusk
I don't think it has to be any more complicated that that unless you have something else written in the contract about reasons for terminating. I would give it to her in the morning rather than afternoon, and on Friday morning, so the child's last day is Friday. Just a thought.