This is a mess and it sucks, I know. You've dealt with your mom's highly-emotional behavior in the past and I'm guessing you, your sister and your father have done quite a bit of egg-shell-walking to keep from upsetting mom. I wish I had a magic word that could drain all the conflict from relationships, but I don't.
This isn't going to come as a surprise to you---you can't change your mother. You just don't have the power. She's in charge of herself and her behavior, not you. You have to take care of yourself (and your kids and husband) and give mom the consequences of her behavior. You have to leave your dad to fend for himself. He's made choices in his relationship with her for years. His life; his choices.
The business with you and your mom is not so not simple.
First, separate out in your mind, your mom's erratic behavior (throwing a fit because you wouldn't bring the kids that day, blowing up your phone machine like an insane headcase, et cetera) from what you contributed to this fight. I'm guessing you were still really mad when you left that message that "if she wouldn't quit being stubborn and listen to [your] concerns, then [you] would stay away from her home".
Come on, who wouldn't be?
Given a rewind button, you might not have said that, but your mom's tantrums aren't new to you. She's obviously run amok before and you've, just as obviously, swallowed a lot, what with not liking conflict and your dad urging you to adopt his placating position. Frankly, most people don't like conflict between themselves and those they care about.
Your making-nice with your mother hasn't changed things. Not really.
Maybe it's time you accepted that her behavior is in her control. You can't change her. You do, however, get to decide that to which you expose your kids, your husband and your self. You probably need to limit your interaction with your mom. Really limit it. Don't tell her what to do, just don't be available for tantrums or abuse. No apologies, no colluding with dad to "keep her happy."
Think of it this way--your mom's behavior has worked in the past. She's got her husband and kids jumping through hoops when she's upset. This isn't good for anyone. It's not made your mom particularly happy, either, even if she calms down at the moment.
You've told your kids that certain behaviors are acceptable and others aren't. Isn't it time you did the same with your mom?
* * *
One of the hardest situations in life confronts us when the people we love make bad choices. Whether this is a spouse who's over-drinking or adult children who don't pay their bills, we sometimes have to face the reality that we can't change others.
Don't think you just need to say the right thing. People respond less to verbal interaction than to changes in your behavior. If you keep yammering about what you don't like, but continue destructive relationships, the message is only that you're a bitch. Others know when you don't want to change things and you'll tolerate crap.
Even when sweet words are spoken, the actions that accompany this speak most loudly.
The hard part is to change you--your part. Look hard and long at what you're contributing to unhappy relationship situations. You can change your behavior very directly. Changing others isn't as easy.