As another teacher, I can tell you're a good parent and that you have a reasonably well behaved child. I have two types of students in my classroom: the kind that are deterred by consequences such as pulling cards, and the kind who do not care. I could go on and on about why this is and why each child responds the way he/she does to consequences in the classroom, but it's not any more relevant than that the two groups exist. A child who is deterred by consequences, or is upset at herself for needing the consequence, tends to be intrinsically well behaved and, though she may make a mistake here and there, values proper behavior. That comes from home, so on behalf of teachers everywhere, thank you. Your child is so well mannered that she didn't back talk even when she knew she was right.
The students in my class have no problem letting me know if I have seen a situation the wrong way. But in turn, they are scrupulously honest about the way they actually behaved or that system does not work. Sometimes there are issues that come up with other teachers in the building or with my aide though, and this is what I tell them to do:
Instead of pulling cards my classroom works on a color system, where they move down a color for infractions in behavior. Each color stands for something: a warning, removal from an activity, referral to the office. If the aide asks a student to change his color, he does so without argument so that the activity can progress without disruption. However, when there is a transition or a chance to talk, the student discusses the incident with the aide at that time and explains himself. The child is much more likely to be taken seriously because he is talking in a respectful way and not causing a scene in the middle of an activity, and it makes him more believable and makes an adult much more open to the conversation. Usually the aide will allow him to switch his color back, but occasionally not. But even if not, I find that most of my students are ok with the situation just for having gotten the chance to explain.
I think that even though maybe some time has passed since the incident, you could have your daughter explain to the aide what happened. You'll have to practice with her so that she doesn't get too nervous, because I realize that she is only 6. If she's more comfortable, have her write a little letter that says something like, "I am very sorry that you thought I was mouthing words last Wednesday while you were talking. I had chapped lips and was licking them, but I'm sure it looked like I was doing something else. I did not mean to upset you." No adult in her right mind could be upset with a child who says THAT to her. That way she's respectfully telling the aide what happened and she may feel more comfortable explaining herself in the future.