@reallyupset11yodd I always had a little bit of a temper, but I was a right little shit around your DD's age, and I would sometimes get violent towards my DM too. I'm not proud of it, but I will admit it here to try to help you understand your DD's perspective.
Basically, I went though puberty quite young and I didn't know how to handle it. I might have done the same in your DD's position. You reminded her of something she did that was bad, her younger DBro immediately did something better than her and even her younger DSis did something that you found acceptable. She felt bad, and her DBro compounded those feelings by pointing out her failure.
The trouble with all the puberty hormones is sometimes they hit before you're emotionally mature enough to understand the impact on your actions on others and to think through things before doing them.
How she acted was unacceptable. But now you know how difficult it is for her to process frustration and rage, you need to phrase things with more sensitivity going forwards, and you need to take care not to let her siblings wind her up so much (hard, when that's all siblings do sometimes). I'm not saying you're to blame at all, just pointing out she can't control her triggers properly yet, and you may be able to help diffuse things whilst she's getting more control.
What really helped me was my DM waiting until I was calm and had thought things through, and telling me that she didn't love my behaviour, but she still loved me. Quite often I wouldn't be able to say I was sorry, but she'd know I was sorry, and we'd both move on, rather than her forcing it. Hearing my DM reassure me she still loved me was really important to me, even if I didn't react particularly lovingly.
I wasn't awful to my teachers. I wasn't awful to my DF or my DBro. I was only awful to my DM, because she the only one I knew in my gut would always, always be there for me and who wouldn't give up on me. I grew out of being a little shit when the hormones settled down, and I remain incredibly close towards my DM as an adult.
Sometimes children can behave badly and it's not a reflection on your parenting skills. I promise you that. Your DD has had a major meltdown and you've had a shock, but this doesn't mean this is who she is now and forever.