I attended a private school from seventh grade to graduation. We had a headmaster and a dress code. Technically, it is an independent, not-for-profit, college preparatory day school. We had sports and extracurricular activities including student government. The school is divided into the Lower School from pre-K to sixth grade and the Upper School from seventh to twelfth grades. Each has its own Principal. We didn't have houses or prefects. What you described sounds more like a boarding school to me. Or Harry Potter's Hogwarts.
