ShirtyGerty - I can see why you're upset with your DH for not believing you. That would make me very cross too. But it does sound like he's happy to talk to MIL to clarify, so job done on that one.
On your MIL, my advice would be to try and focus on the intent behind what she's saying - keen to have a relationship with her DGC, and would like to visit regularly when they arrive - and worry less about the details.
I got very tense with MIL when our DS was due/arrived. He is her first, long awaited and probably only, grandchild. She is still young, but has not been active due to arthritis (she's just had a knee replacement at only 65).
She made lots of remarks about wanting to have the baby visit her, him going to stay overnight, providing childcare, etc, etc.
After the first few months - which we found very difficult as DS was (and still is) a very poor sleeper - I suggested that she might like to come and visit me and DS every two, three or four weeks. It was her choice to come every two weeks (I couldn't face weekly). Initially the visits were quite a chore for me, but after a couple of months where we had some quite honest chats about stuff other than her time with DS, we got into a nice pattern.
As DS is very mobile, and DMIL is not, what has worked well for us is that I leave them to it in the front room while I potter elsewhere in the house - cooking, cleaning, whatever. She does the odd nappy, reads to him, has lovely playtimes.
I've probably only left them twice in about a year of visits. Once to go to GP and another time to go to Post Office. No more than an hour or so. She's also babysat twice for a couple of hours so DH and I could go out.
She has loved her time with DS, and totally accepts that I will leave him with her when I'm ready. But that doesn't stop her building up a relationship with him.
She and I have also got a much closer relationship now, which I value as I am SAHM. She is lovely, so that of course helps!
Despite loveliness, sometimes she says something troubling (like your own MIL). I try to remind myself that I use DH as a sounding board and, as she is single, she doesn't have anybody to fill that role, so her ideas are unfiltered/tested before they got to us. I suspect this might be the case with your MIL's statement.
There is definitely an element of comparing notes between grannies, I would say. But this pales compared to the complete and utter love that 99.9% of grannies feel for their GCs.
Sorry for the length, but hope this might reassure you a little. I'd also go so far as to say don't worry if your DH doesn't raise it with your MIL, she's hardly going to be able to wrench the baby from your arms and leg it, so it will be your rules so far as that goes, whatever she says.