Could it be that Friend thinks DD is too old to need to be included? That it was find when it was go out to lunch w/DD or use lunch $ on babysitter? But now that DD could stay home alone, Friend really wishes she would?
It's not friend's place to decide when OP's DD should stay home alone. Did you miss the part where the OP and her DD already had this outing planned and were inviting the friend along? If she "wishes DD would stay home", she is free to decline the invitation but it would be extremely rude to expect OP to disinvite her daughter.
I find that in these threads some people twist themselves in knots to blame the situation on the child or the child's presence. Take away the fact that she is 13 for a second. If I call and invite you to join my friend Sally and I for lunch, and it's clear that Sally has already been invited, do you get to "wish that Sally would stay home" and make that clear by being rude to Sally at the lunch? Of course not, that's just obnoxious and rude. Your choices at that point are to decline the invitation or accept and be polite to Sally. If you would like to hang out with me and not Sally in the future, you can invite me only. (Which the OP has already explained that she does not bring her DD along uninvited). Really, it's like OP can't win here. People are suggesting that she leave her DD home every single time that she sees this friend, on the off chance that friend doesn't like teenagers, even though friend has never said anything like that. That is just unreasonable.
Friend is being rude, and whether or not she likes teenagers doesn't come into play here. If you don't like teenagers, don't accept invitations when you know teenagers are included. That's on you.
No, I didn't miss the invite setup nor do I think Friend isn't being rude. I still think Friend might have been forgiving of a little kid but now doesn't want to with a teen though. And why she's being rude would impact what I advised the OP to say.
"Forgiving" her for what? Existing? The OP's DD is behaving (obeying her mom) and Friend has complete control over whether to spend time with her, since DD does not come along unless invited and Friend is told in advance if she is invited to join plans OP and DD have together. What is there to forgive?
Forgiving of behavior Friend views as childish. Not the food incidents necessarily, but the wading in the creek or chin on shoulder.
The word forgiving bothers me. There is nothing to forgive. DD has not done anything wrong.
If friend would rather not hang out with DD, that's fine. But friend needs to say no when she is invited to specifically socialize with OP and DD. But to accept an invitation that expressly includes DD and then treat DD so rudely is beyond the pale. OP and her DD have a specific way of communicating with one another. There is no right or wrong involved. If friend doesn't like it, then friend can make the decision to remove herself from the situation or limit interaction with OP and DD