A good example of the issue of whether the most useful definition is general or narrow is Trapezoid, which is a quadrilateral with two sides parallel. Should that be the definition, or should it be ONLY two sides parallel, not four, so excluding parallelograms such as squares?
The Wikipedia discussion of this is good: https://en.wikipedia.org/wiki/Trapezoid#Inclusive_vs_exclusive_definition. The text I'm teaching 7th graders from uses the narrow definition, but I think W. is right that mathematicians do not. Texts do, to simplify teaching. Is that OK? Maybe.
Actually, we could define parallelograms to exclude rectangles, and rectangles to exclude squares, on the same principle of ordinary language simplification. When I think of a rectangle, I don't think of a square.
Ultimately, what definition is good comes to what clarifies thinking. I'm undecided as to which definition does that here.