Hi Banlung,
I'm sorry if my explanation was confusing, I will try to explain it from another approach.
A district is an area of land, usually quite large (the size of a city or larger). Districts usually administer something and have a governing body.
A school district is an area within which all the schools are administered by one governing body. That governing body is usually called a school board.
It would be acceptable to mark the office from which the district is administered with School District as the category. Google does not encourage us to map out the boundaries of the entire district. That building, though, would not be a school; it would be an office that manages many schools.