I would also tell your daughters that this lady is not in charge of them - you are. Such-and-such might be her rule, but they don't live with her. Your rules are such-and-such. Hopefully they will 'get' that. My son HATES to be corrected by strangers, so we have had this conversation because he is old enough now to ride his bike around the neighborhood and I have some neighbors who are just obnoxious and trying to be funny shouting at my son in a joking manner - but he takes it seriously. It was to the point he didn't want to go outside because he knew he would get "yelled at" by this neighbor.
I would continue to be outside when the kids are outside too, and if a polite "she's fine" doesn't do it, then maybe a little firmer, "Ms. X, we have different rules at our house. She is allowed to run. If something happens, I will deal with it, don't worry." Of course said with a nice smile.
My grandma has some outdated ideas too, and when she asks me about the way I do some things, I really am forced to just smile and explain, "this is how we're doing it, it'll be ok, I promise." I have to say it with love, but I HAVE to say it because anything else would be a lie (and I can't lie to my grandma!!)
These are difficult life lessons. For moms and for kids :) Good luck!