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.
Document as much as you can. If possible, you may want to have a neutral but trusted third person in attendance at your meeting with her.
Assuming US here. If she continues to fail to follow basic lab safety procedures, you may have to report her to the equivalent of your university's Environment Health and Safety department which oversees lab safety at many universities or OSHA. They may even have an anonymous tip line. This woman can probably guess that it was you if you try to make an anonymous report given your history, and you may even be fired by your boss if he really likes her more, wants to be vindictive, and you live in an "at will" employment state and lack an employment contract or union collective bargaining agreement if you are unionized.
I think that these things are unlikely, but your personal and labmates' health and safety should probably come before your job and their reaction to it. Depending on your university regulations and state labor laws, you may not be able to be fired for reporting a violation of safety regulations, but if you are in an at-will state, you can be fired for wearing the wrong color shoes or cutting your hair too short, so a smart employer will avoid saying anything that could be conceived of as dismissing you for reporting a safety violation. Keep notes after meetings of what was discussed and email them back to your PI as a record to confirm that you both agree on what was said and what you agreed to do.