I started drawing some lines and curves in CAD to get an approximate understanding of how different points on the reflector redirects the light from the LED.
I draw a vertical line representing the vertical axis of the LED, then two diagonal lines, mirrored across the vertical line, to represent the emission angle of the LED. For an example, for the XRE-R2, the diagonal lines each form a 45 degree angle with the vertical line.
I then draw in a curve, loosely resembling a parabola, symmetric about the vertical axis. Where you cap off this curve determines the depth of the reflector.
The 45 degree lines each represent one of the infinite number of "rays" of light emitted by the LED. You can draw in any ray between the diagonal lines, of course, as long as they hit your curve for this experiment. For simplicity, I consider only the 45 degree lines.
Where they intersect my curve, I draw in a tangent line to the curve, and then a line perpendicular to the tangent line. Using the concept of the angle of incidence, a draw a line that represents the ray of light as reflected by that point on the curve.
Based on my lines and curves, it seems that in general, in order to capture as much light as possible from the emitter and project it into the hotspot, the reflector needs to be deep; and in order to reflect the light rays so that as many of them are as near parallel to the vertical axis as possible, so as to minimize the divergance angle of the beam, the reflector needs to be wide and the tangent lines to the surface of the reflector should be as far as possible from being nearly parallel to the vertical line.
Do I make any sense? Please correct me if I'm wrong.