I think you've handled this really well, and I remember when you first posted about this and it didn't sound like you were assuming that you dd couldn't be at fault at all.
it's absolutely wrong of the other mother to have approached your dd, and it sounds like she and her daughter have fixated on her as being the cause, or at least the symbol, of the girl's difficulties. It's really sad, and it would be a lovely end to it if your dd became really good friends with her and she became a big part of this social group of girls from another school on the bus. But that wasn't that likely before; they seemed to be drifting apart, and it's going to be pretty much impossible now. Your dd is only 13 and while of course she should be civil and kind it's not her job to sort this girl's life out for her, nor yours.
I think it's probably reached the stage where it's better for the girls to have very little interaction now. i also wouldn't speak to the mother until you've had time to talk with dd and come up with some ways forward with her, eg, if she says that she just doesn't feel like they're friends anymore then you can help her come up with for her some ways to be polite with her.
then I would ask first tell the mother that you can see she's upset and worried about her dd having no-one to be a best friend on the bus, because of course no-one would shout at a child at the bus stop unless they were very stressed. It is, however, NOT ok, and that although you won't take it further this time( if that's ok with your dd) it must NEVER happen again.
Tell her what your dd has said about how she sees the relationship, and ask the mother, with that as context, what she would like to happen.
And then just let her speak without interuption for as long as. Sometimes if you don't fuel someone's fire by giving them something to argue against they can come up with more sense than you'd expect.
You can keep very firm that
she is never to approach your dd again
your dd isn't really friends with hers anymore
but other than that, what would she like you to do?
Sorry, i've spent ages trying to come up with something useful, and I suspect this isn't very. I had a similar situation with ds a couple of years ago.
It was harsh at the time, but i think it has been good in the long run. It did make him think about how he acts with people, and actually ended up with real compassion for the other child, after he'd got over the anger and feeling of injustice. I felt very sorry for the kid too; his mum clearly had a lot of ishoos and fears, but everything she did was actually disempowering him more. She gave him no tools to deal with a situation he wasn't happy with, and unwittingly 'rewarded' him for blaming ds, because it fitted in with what she wanted to believe.
Hope it goes ok tonight.