I think it's 6 of one and half a dozen of the other. You both wind each other up, you grate on each other's nerves.
Aside from the snippy emails, which both sides were guilty of sending, if you were better able to communicate with her - and emails are so flat and don't convey subtleties or nuances - I am sure you wouldn't be in this situation.
FWIW, what other word could your SM have used, other than requested? you are getting way too hung up on singular words, and it's winding you up, and so you reply without much grace at all and then she's wound up. If this carries on, you had better cancel those tickets or book a B&B, cos you will kill off any chance to get a nice and pleasant holiday out of this.
You should't have done this on email, you should have called her/your dad to chat about it, sound them out.
Ok so you were wrong, very wrong, and she is milking it for all it is worth, you are gifting this "My Step-Daughter's a nightmare" scenario, even if she is exaggerating it all for effect. You were wrong, you did apologise. A good start, but no where near where you need to get to, and fast.
I admire your comments about what you have said to your DSDs, but fast forward 20 years and they are pitching up with 3 DC for 3 weeks without checking with you first and your DP is in his dotage, and perhaps you will get a hint of why you have put your SM out. The older our parents get, the more set in their ways and routines they are. The less tolerant and patient they are. Forget this at your peril.
As I've said my SDad has been horrific to me, and I've wished a million times he would just vaporise. I HATE the way I have to tiptoe around in what used to be my family home. But it is what it is.
As it goes, the best thing you can do to regain the high ground, and calm everything down is to get on the phone (call for 1p/min here) and say Oh SM, Sorry about the misunderstanding, I just wanted to tell you that we won't be permanently camped at yours, that we are doing X and 'Y and visiting A and B. Again, if you don't do this, your holiday will be doomed, as AF said.
Make sure you WILL help her, make sure you leave everything better than you encounter it. Any less and you will give her room to moan. Don't give her that satisfaction.
You need a huge portion of Humble Pie, and lose this snippy entitlement attitude pronto. I know I'm being harsh, but I am being fair. I assure you, if you don't smooth things over now, it'll be the last chance you have to do so.