Who Dunnit ! - Spoilers

Just watched this remarkable little film, and after reading some of the posts on this board, I went back and reviewed some key scenes. What follows is my take.

In the motel scene, Peter see's the victim walking along with her mother, who threatens to hurt her, and tell her dad, if she doesn't put the stick she's playing with, down. A while later, he hears some screams and crying, then the camera shows someone wearing foul weather gear, (the kind a fisherman would use), and gloves, beating on something with an axe handle or a large wooden dowel, like the one the detective finds in the motel room closet. The one he picks up, is a red herring. It's not actually the murder weapon, but the director wants you to think it is. If it was used to kill the little girl, there probably would be forensic evidence left on it. Peter didn't do it. It was most likely the girls parents. In the movies description though, it says the detective suspected Peter of being a serial killer, but nothing in the movie shows there being any more victims on the island, other than the one little girl.