I honestly cannot see why your mother couldn't have invited mil to your engagement party, it was so mean. No, she absolutely doesn't have to be 'best friends' with mil or see that much of her but leaving her out of such a nice occasion was a massive snub and given how she is now overreacting, incredibly hypocritical.
I agree with thid. It was horrible of your mum and TBH. I would have insisted my MIL came, otherwise she can throw the party in my absence.
My parents paid for our wedding, but thst didn't mean they could invite the world and his wife. We gave each set of parents an allocation of invites and they could invite who they wanted within that number.
I didn't want the wedding overrun with old folk and wanted most guests to be friends of me and DH.
I'm Also from a culture where MILS would be friendly upon marriage, so I can't imagine my mum being so unfriendly towards MIL.
FWIW, my mum also had people asking why they weren't invited to my wedding and they'd heard so many people talking about the upcoming wedding.
She told them thete was a limited number ab it's not like 'back home' where everyone can attend.
Your mum is being unreasonable.