If I refer to her as моя подруга, would that send the wrong message?
Yes, it would.
If I introduce her to other Russian people, do I refer to her as мой друг?
It's better, however some still can understand you wrong, but why should others care? If you in her presence begin to explain away - "she's just a friend not more" - it does not look nice.
You also can say: Она друг нашей семьи (She is a family's friend).