I don’t think there’s any issue over the frequency of visits- it is what it is, and it’s entirely up to you.
I think your problem seems to be the ‘flying monkeys’ that report back to you your MIL’s complaints about you.
You can’t stop your MIL bitching about you, and on past form it’s actually likely she’ll indulge herself often. Next it’ll be that you’re not feeding the baby enough or have a poor sleep routine or dress them in a way she hates. You can however put in place boundaries with family about the reporting back. Be clear once they start up about MIL, that you’re not interested in second-hand commentary on your parenting. That if anyone has anything to say, they can say it directly. Be firm and repeat often, till everyone is bored with the game.
I know it’s very difficult not to take offence when you hear what she’s been spreading, new mums are especially sensitive about this stuff. It’s natural. It certainly will dampen your enthusiasm for doing her favours.
Know that grandparents often complain about not seeing the baby enough- even the ones that have good relationships with the parents. It seems to be a way to say ‘I love the baby so much’ with a side order of martyrdom thrown in! In fact the ones I’ve known to say it the most are the ones who go on to want little involvement (and offer no real help) in the child’s life down the track! Involved, mature grandparents will find ways to interact and help out, even whole continents away.
Feel free to let your MIL set the pace of the relationship with your baby. If she puts in little to no effort, and spreads ill-will, then get on with your own life and let her reap what she sows.
I dealt with nonsense like this early on in my baby’s life. I flat out asked (in a conversational tone, face to face during our next visit) ‘Did you tell xx that I xx?’. The bluster and awkwardness I caused her by being direct soon put her off. And she knew after a while that she had a choice. Behave or be in the background.
Congrats on your baby 