You can't just tell the other parent that your babysitter's services are available to mind their child as a favour without telling the babysitter in advance and asking her if that is ok with her, and setting up a payment arrangement. The other child's parent got free childcare from someone who makes her living or part it if anyway minding children.
How exactly did you word the text in advance that you sent her -- was it a case of 'I have arranged that X's child will be there with DD this afternoon as X has to work late; is this ok with you?' i.e. did she really have the option to refuse and thereby throw a spanner in the works that you had already set up? Did what you said strongly suggest that this was a done deal?
She is not your employee. You can't do that to someone who just has an hourly arrangement with you.
I used to work as a nanny and I would have expected you to offer that parent X should get in touch with me by text or whatever and work out a fee for the childcare. Either that or offer to pay me to take care of the extra child.
Like RattieOfCatan, I suspect she is fed up of parents arranging playdates or piling on extra kids and taking the piss. Or she may be worried about consequences for her job if a child she has never met before turns out to be a bolter or someone who goes wild in someone else's house, and ends up hurting herself or someone else or causing damage.
I also suspect the other child was a handful and she felt she had earned the money she asked for. Your child has bad asthma and on top of the responsibility of that you arranged for her to have another child there who may have wound her up. Two of my DDs have asthma and I would never have had another child over to play or to mind if they were going through a rough patch. I think this was poor judgement on your part.
It really doesn't matter that you're home early, you've booked her until X time so you pay until X time. She can't book anybody else in for that time even if you are home early. It is nice to let your childcarers go early if/when you can. We do appreciate it, but equally, you aren't doing us a "favour" by paying us for the time you booked but didn't use.
I agree 100% with this too.
You have my sympathy, Rattie.
My DDs babysit a lot and wouldn't go back if they had ended up looking after more kids than stated more then once in the same house, without any offer to raise the hourly pay.
In sum, YABU.