I guess I'm in the minority but I think the fault is with Lostindlewonder. I've double booked and made other mistakes, and mistakes really piss people off. You would think that just because it's piano lessons that people wouldn't get so upset. But that's definitely not the case because people are just so wrapped up in their own lives\schedules\feelings. And it's upsetting when there is a bump in the road.
I know it sucks to hear, but you're in a service business. Not a tit-for-tat busines. Just because a client asks for an exception\pass does not mean that client will happily grant the service provider an exception\pass. They won't. It's easy think that because you're in direct contact with these people on a weekly basis.
In the service business, you can't make mistakes. Period. Each mistake costs you a client and if you're lucky enough to have a client who overlooks one or even two mistakes, you better be grateful. I think you're getting caught up in how people should treat you respectfully, ect. And it's true, people should. But you are a business providing a service (not being the neighbor across the street). People want what they want, and they want it now. AKA, the customer is always (usually) right.
Were I in your situation I would have apologized profusely, silently cursed them for not understanding that I'm not perfect and make mistakes too, and then offered some type of credit to appease them. And going over your description, if it's accurate, it doesn't seem as if they treated you that badly. You called them. They expressed their feelings in a respectful way, even though it hurt your feelings, and then things escalated from there. Your clients could have responded in a multitude of ways including pretending to be fine then secretly going behind your back and getting a new teacher. Instead they were direct. I'd take on a studio full of direct students today.
Oh, and one more thing, if your actions went against your studio policy, then you only have one person to blame. If you can't enforce your studio policy, then get another one. Don't feel like you have to have the same policies as everyone else. Use what works for yourself. I mean, if you have no problem rescheduling students, then do so. Just have a system in place that ensures you don't double book. For example, instead of giving the student a time right then and there, get back to them at the end of the day after you've triple checked your schedule.
Ah, and last thing. Promise. When clients or customers get upset, there is usually a learning moment somewhere. Always go back and review everything and ask yourself if there is any one thing, no matter how small, that you could have done to have prevented the situation. If there is one thing, like not double booking, then you can't place the blame anywhere but yourself.
Just a humble opinion from someone who's made a lot of mistakes over the years.
Topic: Students who think they can treat you badly.