I had a doula and my sister at my births (as well as my husband). It was really lovely to have the emotional support - for both of us. We were in agreement about it, and I had long labours, so it was great for them to be able to tag team a bit.
I think it's a very individual choice though. I love my mum, but wouldn't have wanted her at the births, as she had very stressful births with all of us, and would have worried if anything didn't go according to plan.
She looked after my 1st and then my 2nd whilst subsequent children were born, and was with me in early labour for both, which was great.
I'm telling you this to counteract the pp who have all said it's weird to have anyone other than your partner at the birth.
That being said, it is such a personal choice. The people I had at the birth were all very respectful of our choices and melted away once the babies were born, to enable it to be just me and OH getting to know our new arrivals.
If you feel judged or stressed by someone at your birth you don't need them there. It sounds as though your mum will not add support in this situation.
Remember, all the way through it's your body and your decision. It's an intense and personal time, so anyone who makes you feel bad about your choices is not the person to be there.
Good luck for your induction, and don't worry about your mum. When you text to say that the baby's here she will be over asap, if nothing else to make sure she gets a cuddle before MIL!