A.S.
What your SIL did is what I call "carrying tales."
Anytime someone "carries tales" to me, I tend to look negatively at the tale-bearer rather than the original tell-er if that makes any sense. Most of the time people carrying tales are up to no good. In other words, they are looking to one-up someone else, or garner favor from a parent or other authority, etc. Every now and then someone will "carry tales" for a legitimate reason - i.e., to warn someone to protect themselves against another person's wrong-doing.
The way you presented it - it sounds like your SIL was "carrying tales" for the wrong reason. She wasn't trying to protect your mom - she was trying to hurt you. And your sisters.
If I were you I would get my mom and sisters together and say "look, are we really going to allow this interloper to come between us after all these years ? Especially when we all know the trouble she has caused before? Mom- I have a feeling that whatever she told you that I said was blown WAY out of proportion and was not offered in the context in which I presented it."
I'm not saying that what you did was right - but the bottom line is what your SIL was WAY more wrong imho. She escalated the situation and breached your trust. I would never, ever trust her again.
Good luck and I hope all this works out for you.
PS: I highly doubt confronting her will do much good but if she asks you why you are giving her the cold shoulder then I would be honest. "You tried to destroy the relationship between my mother and me for no good reason, and unless you sincerely apologize and make amends it's going to be hard for me to move past this."