So you should explain to her the 3 different types of stutter a person can have. Repetitions is what your showing with the I-i-i was at the-the-the park. There is also blocks where words just won't come out and you are just unable to speak. Then you have prolongation where you might be saying i have to tell my mmmmmmmmmom about school. Now everyone doesn't stutter the same way. I've met a few people who have stuttered and we all sounded different. I have a lot more blocks and some single syllable repetitions while my friend is stuck on prolonging all his words. You should also make her understand what can cause a stutter. With a change of emotions can cause a change in fluency. I get it whenever i feel anxiety and sometimes excitement. I could be talking to my friends about a new game and just stumble on every word. If she still insists that you are faking it then talk to her. Legit monologue to give her a whole essay of examples of your stutter. Have her ask your family about your stutter. She can't deny it if multiple people tell her about it. And if all that fails then maybe she just doesn't want to take your disability seriously. You should have a serious talk about your relationship at that point.