In this case, I think you need to ignore everything SIL said except, "you are welcome for Christmas dinner. We want to move ahead". The rest is noise.
Go to the dinner, treat her with kindness and be friendly and let their response and behavior be your guide.
Someone mentioned that SIL ignoring you at your own parties may have been embarrassment. That's very possible.
I get what people are saying about your children watching and allowing them to treat you with disrespect. But, if I read things correctly, mostly she's just ignored you. Other than calling you crazy, which was probably a defensive reaction to being called out, it doesn't sound like she's going around spreading rumors.
Since she/they've expressed willingness to try, I would let them. If they continue to act the same way they did before, then you can react accordingly.