She was angry about it because she was ashamed. She'd been trying to make it go away by scribbling it out and you reminded her.
Rightly, I might add.
I'd go down and make friends. Apologise for shouting at her and over her. You can't be judgy on the way her anger manifests itself when yours was crappy too. Acknowledge that - you didn't handle it properly, you let anger get the better of you, you're sorry. Do you model saying sorry when you're wrong in your house? Because that's the start of her feeling comfortable with saying sorry, that it's ok to acknowledge your mistakes, we all make mistakes and say things we shouldn't - but it's important to make up for it.
My husband joined our family when my son was 6 and at first he was deeply sceptical about saying sorry for how you respond to a child's outbursts - because they 'started it'. My view is that we are the adults, it doesn't matter what led to the altercation - always get in there and say sorry if you didn't handle it well, because eventually, they'll start doing that too.
I feel fully vindicated in this approach because my now adult son freely, abundantly and sincerely apologises if he has crossed a line. We have modelled the right thing to do.
ANYWAY. Go and say sorry. Tell her that you love her. That you want to move on from this as much as she does, and that it really isn't that big a deal - because it really isn't - but she does need to say sorry, just like you have done. Because when we upset someone - especially someone we love - we say sorry.
You seem to default to kneejerk punishments in the moment, rather than rational consequences. "We had a heated exchange so I took the TV remote away." That makes no sense - you're just exerting your control over her, and she will resent it. It's not discipline. It's punishment.
Here, that kind of behaviour would lead to a withdrawal of co-operation. We need to work together, we're a team, we're on the same side, I am not the enemy - but while you're treating me like I am, it makes me less inclined to want to co-operate and work as a team with you. We can either help each other out, treat each other like civilised human beings, or we can carry on like this - it's up to you.
Then the next time I got a 'mum can you give me a lift to....?' I would remind her of the co-operation/same team thing.