The reason is because you want to be able to usefully talk about 'true vertices'. Another way to define a polygon is any convex hull of finitely many points in Euclidean 2-space with nonzero area. Then the 'real' vertices are those points on the boundary curve that are nondifferentiable ('pointy' or 'sharp').
Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.
Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.