Number of distinct enclosing circles for N points
An enclosing circle (by your definition) corresponds to a triangular face of the farthest point Delaunay triangulation, which is a triangulation of the convex hull of the input. For points in general position it always has exactly $h-2$ faces where $h$ is the number of convex hull vertices. Therefore the number of enclosing circles is also $h-2$.

They can be generated by constructing the farthest point Delaunay triangulation, in $O(n\log n)$ time.