My daugher attends a Montessori preschool and they sing the song the same traditional way that you've probably taught your child. Even if it is different I don't think you are messing up your child.
Kids are smart, they'll just learn a new version if that's the case.
If you feel unsure still about what your area school teach, you could call a local school and see what their thoughts are. They don't know who you are, so there shouldn't be any embarrasment in asking. My daugher attends a Montessori preschool and they sing the song the same traditional way that you've probably taught your child. Even if it is different I don't think you are messing up your child. Kids are smart, they'll just learn a new version if that's the case.