What's her name? Is it something other people have had trouble saying before?
I don't think you are overreacting. In my opinion, she's either not paying attention, doesn't care or is using it as a power play in some way. Particularly when said to customers, it's disrespectful. None of which are ideal things for a new employee to be doing.
I'd get her boss (your mom) not you. to give her a strict rule on it. And a definite time to be 100% error free by. If she doesn't see the seriousness of it by then, then bye bye.
It is one that isn't super common, and people can mess it up the first time or two, but no one has consistently messed it up for almost three weeks. She keeps using names like Lorraine and Eleanor, neither of which are the actual name.
To me it screams inattention to detail and flat out not caring enough about your job. as
@Despairagus said earlier, knowing the name of your boss is kind of crucial and is a red flag.
My heart is sinking as I don't want to go through another hiring process and she has some good points, but there are a few red flags that make me think we might soon be seeking another employee (e.g., we have told her several times to NOT take any actions on work orders or sending people out without checking with us first since she is new, still in training, and we don't do everything exactly the same every time and until she has been here a little longer please run things past us first so she can learn. She cannot seem to do that. I do admire her willingness to work independently, but honestly, she has to CHECK with us as every client could have a different scenario.)