H.W.
I think you are the one overstepping if you asked the sitter to come to you and to tell her other client that she 'needs to check with you first'.
That's highly unprofessional. She is a sitter. She has worked for you for a couple years, which is great, right? Why make her uncomfortable by putting the sitter in the middle of this? Your friend may very well have plans that far in advance. Maybe, just maybe, she and her husband decided they needed regular weekend 'date nights' and put it on the calendar? Why is this your business?
I think the other thing that raises my eyebrows is the attitude that you are entitled to 'first dibs' on another human being's services. I was a nanny and did childcare for years. *I* conducted my business, not my clients. I worked the days they scheduled me for and what I did with my other time was my business. If you want to keep the babysitter feeling comfortable about the situation, then I would suggest backing off and not putting her in the middle. Instead, the appropriate question is 'what's your availability for the week of xyz..." and go from there.