What can I do if a student in my lab threatens to tell lies about me, and my advisor doesn't want to fire her?
She has been mean to you, not the PI. He clearly signals that he likes you and your work. You say that he also realises she has an immature attitude. I think that you are in a good position here, unless you mess it up. 

The superviser clearly sees the situation and - in veiled form - tells you to be the mature part of the interaction. When he tells you to forgive her, that's very clearly what he means. In short, at this stage, you have to show maturity. Clearly, you need to work on that (I quote your response: "Hahaha...." - SE is not the place for such formulations).

You should not worry too much to prove that she lies about you, as the superviser made very clear that he will not pay heed to her stories about you (why otherwise would he have emphasised that you are one of the brightest students he had?). He has sent you a clear message of confidence. 

Whether he likes her or not, you do not know, but it seems very clear, even through your report, that he takes pity on her. He will not fire her; it is not your job to make him do so. He also will not remove you, unless you decide to go. He seems a very reasonable person.

As for her using threatening language, you have a number of options: when she does that, you ignore it. Or else, you find a reason for yourself to go away, without responding, to some made-up excuse (for yourself), such as the bathroom. Or else, you respond: "This is not the place for threatening language. Please refrain from such language, it is inappropriate to talk like that to your colleagues." - or something along those lines. In a cool, calm, and collected manner. If she indeed has the maturity of a 12-year old, she may not absorb the words, but she will absorb the tone.

She presses the buttons where she sees an effect in. If you signal to her that you have no business with her, and that you are unaffected by her threats, you will become an uninteresting target. 

Forget about winning/losing. You just do not have to play her game, you decide which games you play, and which ones you don't.

Bottom line: stay away from her; relax, the PI trusts you; and follow your maturity level and do not let yourself be dragged down to hers. She is not the one to set the agenda, your PI is, and he unambiguously signalled his trust in you. When he sets the meeting, if she apologises, graciously accept the apology (even if you do not believe in it, take it at face value), but still stay away. Her safety procedures are not your business anymore - you have made your case.