Is the title, "managing director" used in American or European nonprofit organizations, specifically in elementary or high schools? And if so, does that person stand below, equal to or above the principal?
In Britain, most independent schools and some state schools have a bursar, whose remit is the finances rather than the educational side. Whether the bursar is senior to the head is a very thorny question; in theory they are both supreme in their own sphere, and the spheres do not intersect. Wise school governors do not press the point.
Managing director is not a common title for a school official in the United States, though it does exist.
In a traditional public school (i.e. government-funded and government-managed), the principal is both the chief administrative and chief pedagogical official, and usually overseen by the superintendent of schools (or simply superintendent for short) who is the chief executive for the school district, hired by the school board.
There is considerable variation outside of this model: there are all manner of charter schools (government-funded, independently managed), parochial schools (church-funded, church-managed or independently managed), and various other non-profit and for-profit private schools. The head of such a school may be a principal but headmaster or headmistress is not uncommon, after the British fashion.
In a parochial school, oversight of the head may lie in the ecclesiastical hierarchy, e.g. a pastor will have ultimate oversight over a parochial school, or a Catholic diocese or religious order may appoint a commissioner or superintendent for their schools. Otherwise, like other private schools and charter schools, the school is commonly overseen by a board whose members may be known as directors, trustees, rectors, governors, and so on.
Where a member of that board possesses executive powers and duties, he or she may be known as the managing director or executive director, and may be analogous to a superintendent. In an independent private school, particularly, educational/pedagogical and administrative/financial functions may be split, with the latter being a managing director or executive director, but also president or other titles. I attended a high school operated by a Catholic religious institute; we had both a principal and a president.
I would not say that school director, managing director, or executive director are common. A web search turns it up mostly in charter school networks and in newer, non-sectarian private school systems, whether non-profit or for-profit. I surmise they choose that title over superintendent or commissioner specifically because it is nontraditional and exudes a business-oriented or technocratic approach to education. Parents dissatisfied with traditional public schools and considering other options might find that appealing.