It is great that you have a good relationship with your mother!
Because of her experience with being falsely diagnosed, I can see why you are hesitant to tell her of your own experience. Some questions for you to consider: By not telling her, will your diagnosis and treatment become a wall between you and her? Could your hesitancy to tell her be because of your own hesitancy in accepting your bipolar diagnosis? Looking deep at our feelings in telling others about our being bipolar may reveal some of our issues with our diagnosis.
Good luck!