There's a big difference between wanting to see what the forum is like and looking through everything she has ever posted on it and then spreading gossip about it through the family.
You may as well admit you were out to cause trouble for her intentionally because you've already admitted you are glad you have caused all this bad feeling though out the entire family.
Otherwise you could have ignored it, spoken directly to her about it, or confided in your DH and let him decide whether to tell his family or not.
And it's not just her you have caused trouble for, it's the entire family. Her husband (your DH's brother?) will now have to chose between supporting his wife and appeasing his family. Her children will know something is going on and no matter what the reason they will just be upset that everyone is being nasty to their Mum. They won't thank you for this and think badly of her, it's you that they will blame and perhaps that will strain their relationship with your children. So all the cousins will lose out because you wanted to get one over on her.
What if they feel they can't visit anyone at Christmas now? Why should her children be excluded just because you wanted to expose her as a nasty piece of work while being one yourself? What if they do visit and the whole family ends up arguing? Your children and hers could be in the middle of that and no matter what you think, YOU will bear some/most of the responsibility for it.
Does your DH deserve to to fall out with his brother? Because that could happen.
And when the whole family are utterly sick of the entire situation you best hope they don't decide to shoot the messenger and blame you for sticking your nose in and not keeping your mouth shut about what you read.
She might be a liar but you sound like a trouble-causer who enjoys causing upset just for the sake of it.
There's a big difference between you overhearing someone talking about you and you deliberately seeking out posts by her and then tale-telling on any little thing she has said about someone else.
No matter what your SIL said on that other forum, she's still managing to come off as the better person when compared to the trouble you have caused for no good reason but to "make people know her for who she really is."