My 2 cents
I think you should have canceled having the other woman over. She is there every week, your mother is only there for two! My MIL comes to our home every Wednesday to do bible study with my son, when my mom is town she graciously backs away and lets us spend time with my mom.
My other point is I think you set your daughter up. You know your mom, you know she is easily offend, but yet you sent her in there to tell Grandma, "I don't want to spend the day with you I would rather stay and be with so-in-so that is here all the time." I'm sure your daughter worded it better then that, but who wouldn't be offended! I'd be crushed if my grandchild did that and I'm not easily offended!
I think you owe both your mother and daughter an apology.