I would ask mil again about her comment re no contact, challenge her on it.
It's fairly obvious what happened, though, isn't it?
MIL used "well in that case we had better not see each other" as a piece of theatre, assuming that the OP and her husband would immediately see the error of their ways. They didn't, so she was a bit stuck. She clearly hasn't told her husband that because it makes her look a bit of an idiot so has presumably spun some line about OP being hard work, difficult, leading son astray, cutting them off for no reason, etc. FIL, whom one suspects would not score highly on tests for emotional intelligence, thinks he's got to defend poor misunderstood MIL from nasty irrational DIL and son. MIL won't admit that she's been an idiot, and by the sounds of it doesn't do apology. Impasse.
The MIL could, presumably, break the logjam at any point by phoning or writing to the OP and her husband, showing some insight about the discussion which culminated in her saying they should go non-contact (new readers should recall that the non-contact was the MIL's idea, not the OP and her husband's). If the MIL wrote in reasonably conciliatory terms, I think the OP would have to be very hard-hearted to not at least give her a fair hearing. Instead, the MIL won't be honest with her son and daughter-in-law and won't be honest with her husband, so the current situation ensues. If the OP's family actually wanted to see their epistolatory relatives it would be different, but they aren't particularly bothered.
The OP's MIL is probably over on gransnet now, hunning and loving on the unhinged "cut out of their lives" threads which just leave you thinking, "Christ, if I had relatives like that, I'd have cut them out of my life a lot sooner".