I treated a lap dancer a few years ago who had fallen off a table while performing and ended up having a partial meniscectomy and a Wikipedia reference-linkrotator cuff repair.

If you're really embarrassed about how you injured your knee, tell them you slipped on the stairs - your treatment is not dependant on how you injured the knee, it's dependant on what's wrong with the knee.

Doctors and physios have all worked in hospitals and I guarantee that your story would be tame in comparison to many that they will have heard. Get it sorted!