I'm really failing to see why people think the OP is so awful here.
Okay so maybe things could have been handled better but I imagine having someone follow you around your house (child or not) verbally abusing you to the absolute end of your tether would cause you to lose rational thinking for a while and want them out. And yes I'd think the same if it were my husband (we'd all be telling her to get him to leave if it were him).
I read it that when OP said she 'didn't care' where she went it was in the heat of the moment and once she'd calmed down not long after she arranged for her daughter to stay somewhere safe or would have brought her home if not.
It's hardly the worst thing a parent has done is it. I'm sure my parents would have done something similar if I'd told my mother she was a cunt who I wished would die following her around the house getting in her face.
She's nearly 18 for Christ's sake. It's not okay for her to be verbally abusive to anyone, not even her mother. It's a disgusting way to speak to your parents and I'm not surprised op lost her rag.
All of this 'its just what teenagers do' stuff is nonsense. I would never have spoken to either of my parents this way because my behavior wouldn't have been excused and there would have been consequences as there should be.
She was told to leave her house for one night after being utterly vile towards her mother. OP has said she arranged for somewhere for her daughter to stay or would have let her home if not. I really don't think this counts for 'piss poor parenting'.
OP now things have calmed down I'd try and take this opportunity to sit down with your DD2 and get to the bottom of this behavior, try not to let it escalate into an argument again. If this is a regular occurrence would you be able to look at some form of anger management perhaps for your DD?