This helped me understand what A TIR is and how the different types work. Great stuff for those of use that are trying to understand and learn how things work 
That’s a great guide.
Something that readers might not know by from that document (although the comparison to reflectors shows how it works) is that a TIR is useful for blending the light output from emitters that have bad tint-shift in reflectors. I’m mostly looking at Cree here for examples; the XP-G3 is nearly unusable in a reflector but tolerable in normal TIRs, and in a broad-angle (45 or 60 degree maybe) pebbled TIR it’s perfectly fine. In fact my Klarus Mini-One Ti uses XP-G3 behind a pebbled TIR and I have no complaints.