S. -
First, I'm really sorry that this happened to your daughter. However, I'm not sure I would approach it the same way you are. For one thing, I don't think one incident like this will label your daughter as a troublemaker. And stuff like this happens...I am sure I have made mistakes with my own two children where I misinterpreted what I saw or heard. Having spoken to several teachers over time, I suspect that you will risk labeling yourself as an unsupportive mom who will always side with your child and not the teacher if you go in all upset about this after the first day. The teachers, in general, have a really hard time maintaining discipline in the classroom, and one of their biggest issues is parents who will never believe that their child did anything wrong. I believe you that you know your daughter and she probably told you the truth, but my point is that the librarian doesn't know either of you.
My suggestion? Let your daughter know that you believe her. Tell her that the librarian probably made an honest mistake, and that you're very sorry it happened. You can even tell her that the librarian probably didn't mean to make a mistake like that and would probably be sorry if she realized that she had (in other words, the librarian is not necessarily a Mean Lady). Let your daughter know that life sometimes isn't fair. Let her know that she did the right thing by telling you, that you're there for her, and that if problems continue you will talk to the librarian. And also let her know that, even if the librarian makes a mistake, she is still in authority and should be obeyed. I know that is hard to tell her, and a hard lesson for her to learn, but it will help her to NOT be labeled a troublemaker if she learns to respect authority even when she believes they are wrong...after all, many times she will be the one that is mistaken due simply to her inexperience.
And then, talk about the good times she had at school and encourage your daughter to go back and enjoy it.
Note: I'm not saying to never talk to the librarian about it. Maybe in a few days it can be mentioned. And PARTICULARLY if there are continuing problems, of course go to talk to her (and, if that doesn't work, talk to her supervisor). But if the librarian feels immediately attacked over one single incident on the first day of school, that could create a LOT more problems than it is worth.