When I was growing up, I learned that a trapezoid has exactly one pair of parallel sides, which means a trapezoid is not a parallelogram, so clearly there is a major difference of opinion on the definition. Ever since I began teaching Course II, NYS has been quite clear on the definition, which doesn't necessarily make them the authority on the subject but at least it provides some level of consistency.
After all, the area of a triangle has the same formula as the area of a trapezoid and parallelogram.
I also studied the properties of a kite, which doesn't seem to exist anymore, which is a shame because it offers so many great counterexamples to misconceptions about quadrilaterals.
Steve Goldman Half Hollow Hills HS East Dix Hills, NY