*I’m a bit bemused by some comments here.
Do any 13 year old’s really need it explained to them that swearing at your mother, never mind aggressively, in any context- is unacceptable? Let alone needing to explain why it would be so. I’d be very surprised if a 13 year old did not know that it was wrong to swear at their mother, and why.
As for “just accept her apology”. Why? I would not accept an apology which basically said “you deserved being spoken to like that”. It’s not an apology, for a start. It’s also virtually never acceptable to aggressively swear at anybody. So she is either not sorry at all and thinks she behaved reasonably, or is too stubborn to admit it. Neither would be acceptable in my home. A proper apology would be accepted, anything less would not. Why should I accept an insincere apology? That’s disrespectful- apologise meaningfully, or don’t if you don’t believe you are in the wrong. But then you must accept the consequences of that choice. Don’t insult me with a non-apology and expect the issue to go away like it never happened. What does that teach them?
I think OP is right to raise concerns about this friendship, if she has them. At her daughter’s age, I think it entirely appropriate for a parent to say that they think a friendship isn’t working well/a friend isn’t behaving nicely or any other concerns they have. OP has said that it came out wrong, and has already apologised for that- but do some parents honestly believe they can’t say anything negative about their children’s friendships “in case it upsets them”? I have vastly more experience than my children- if I notice an issue, I will raise it with them. There are very few situations where I would try and put a stop to a friendship, but if I see something is not right/they aren’t being treated well/friendship is bringing out the worst in both parties etc- then I reserve the right to say so. They might not like it, but at 13 years old I’m afraid I don’t think they should just be given free rein when it comes to friendships- I think it’s perfectly fine for a parent to say “I don’t think this friendship is good for you....”. I would expect that my DC might not be happy that I expressed concern/disapproval, but I would expect them to keep a civil tongue in their head at all times. I don’t swear at them, or anyone else in the household. I expect the same from all members of the household in return.
“Taking away their Wi-fi will just make them resentful, so I wouldn’t do that”. What? God forbid a teen should be unhappy about a negative consequence for bad behaviour! Do you only give punishments that your children won’t be put up nor down by? If so, what is the point? I’d look on it as a salient reminder that privileges are not a right. Why on earth should I give privileges, luxuries or put myself out by giving lifts etc to a teen who is rude , aggressive and disrespectful towards me? I wouldn’t tolerate that sort of behaviour from a partner, adult family member, friend or work colleague, so I’m certainly not going to tolerate it from a teen DC. It’s not simply a punishment, it’s a consequence for unacceptable conduct. Children, especially younger teens, need to understand that their actions have consequences.*
You don't have teenagers do you, nor do you appear to have any actual understanding of teens brain rewiring so that in a lot of aspects they basically become giant toddlers again. You're in for a bit of a shock 