Can I imagine a stereographic projection as looking on a reflecting sphere from an infinite distance?
Or could I say that it's the same as a perspective projection but the imaging plane is a sphere which is then parallel projected onto a plane?
The first isn't even approximately right: looking at a reflecting sphere will cram all directions into a finite circle in your field of view, and compresses things far from the axis of vision, whereas stereographic projection stretches out to infinity and stretches areas far from the axis.
The second suggestion is closer, but still not entirely right.
An exact analogy would be something like: First imagine taking a picture that projects onto a sphere, like a 360×180 degree panorama, or a full-surround Imax theater. This spherical image is designed to look natural when observed from the center of the sphere. Now assume that without changing the spherical picture we place ourselves right at the edge of the sphere, then point a camera with ordinary perspective optics back towards the center and take a picture. The result will be in stereographic projection.
This description, however, obscures the most distinctive feature of the stereographic projection, which is that it is conformal, i.e., it preserves angles. If we have a small area anywhere on a stereographic image, it will be exactly similar (except for scale) to the picture we could take with a high-zoom ordinary lens pointing in the appropriate direction. In contrast, ordinary perspective distorts shapes far from the axis of projection.