MyNewNickname gives many great tips. A good strategy in any situation like this is to get the other person to talk first. You only have your daughter's viewpoint, and kids are sometimes reliable reporters and sometimes are not. I'm not saying she's lying - I'm saying she may not know the whole story and she, naturally, has a kid's viewpoint.
Don't lay out your whole argument as you have here. I'm not saying you're wrong, but I'm saying you put yourself at a disadvantage if you make her defensive. Do say, as suggested below, that there was an incident and your daughter was extremely upset, and can the teacher explain what happened. You can also say at some point that you understand there may be information about other children that cannot be shared with you. (For example, if she patted a child who has special needs of some sort, that may be a signal they have worked out together for a special reason.)
Then sit back and listen - without listening in order to respond. Add in, as needed, "Can you explain that a little more?" or "Would you elaborate a bit - I'm not sure I have the whole picture." You can also pose questions like, "Since Jane felt that XYZ, can you suggest ways we might work together to help her?"
I wouldn't bring up the prior incident right away. But you can add, "There was a prior occurrence where you felt Jane was not honest about an assignment? Can you fill me in on that?" Again, let the teacher talk, but you can say - as non-defensively as possible, which is really hard - "Are you telling me that you think Jane lies?" Again, don't start there, but you can wind up there.
At the end, summarize - and write stuff down: "So, to review, you are saying that Jane is not truthful, and that reprimanding publicly is an effective strategy?"
If you still have concerns that the teacher isn't doing right by your daughter, talk to the principal or school psychologist.
At 12, it's also important to teach your daughter how to interact with her teachers and how to speak up respectfully but determinedly on her own behalf. She can also go back to the teacher later and say, "I want to talk to you about the incident the other day. I really feel that there's more to the story that you didn't fully understand. Perhaps I didn't explain it fully and I'd like to do so."