There are two versions for any figure in plane geometry: synthetic (i.e., Euclidean) and analytic (i.e., Cartesian). These two domains differ in how they relate boundary cases to the other cases, but the boundary cases themselves are treated the same in both domains. The synthetic version promulgates exclusivity, rejecting the boundary cases (e.g., “A parallelogram is not a trapezoid.”), whereas the analytic version accepts the boundary cases (e.g., the Trapezoidal Rule (for approximate numerical evaluation of integrals) accepts rectangles as trapezoids). If you don’t make this distinction between the synthetic and analytic points of view, you get bogged down in an endless debate about the merits of ‘exclusive’ versus ‘inclusive’ definitions.
The reason that elementary school champions the synthetic view, while secondary school champion the analytic view corresponds to their historical development: the synthetic view, dating from antiquity, is more natural and intuitive. The analytic view, which came much later, is a quantum leap in sophistication.