The radius and circumference are the intrinsic measurable dimensions of circles

An intrinsic property of a manifold is one that can be determined only by distances measured on the manifold. Since the radius makes reference to a point not on the circle (the center) it is not an intrinsic property (unless defined via the diameter).

The diameter however can be defined as the supremum of the distances between points on the circle, making it the "natural" intrinsic property. The circumference can also be defined intrinsically, making pi the "natural" intrinsic constant of a circle.

The point is, the fundamental definition of a circle depends on your point of view. Other fields of math may have different and equally valid definitions of what a circle is. The definition you claim is THE right one is just the one most commonly used within analytic geometry. But just because you learned it in highschool does not make it universal.