Apparently 1/8th output impedance (or less) is best for damping, etc. But how do you determine what your equipment has with the inconsistent specs that are provided?
Slippery explained that well in post#6 but in case you are still confused, basically that 1/8th rule mentioned in the article is about the "damping factor", not "output impedance". DF is defined as the ratio of the load impedance (the loudspeaker in this case) to the output impedance of the source (amplifier in this case). The 1/8 mentioned is the inverse ratio of DF, that is the ratio of the output impedance of the headphone amp to the load impedance.
The "1/8th rule" referred to in the article HD linked simply meant as long as the DF (damping factor) is not less than 8, you are good. So it was meant to be the more like a minimum requirement, but not necessarily the "best".
It can be shown by calculation that, due to the inherent impedance of the signal path (e.g. wires) between the amplifier output and the speaker binding posts, DF greater than 20 will offer negligible benefits, though much higher values would likely make some audiophiles feel better.
My best sounding amp (on speakers) states a headphone output of 150 mV @ 8 Ohms. How does that all translate?!?!?
All it tells you is that it can give you 150 mV even when connected to a very low impedance (for headphones) of 8 ohms. It will most likely be capable of producing much higher output if connected to a headphone that has higher impedance, say 300 ohms, but you cannot tell from that simple spec what the maximum value will be. A headphone amp is inherently limited in both output voltage and current, and its maximum clean output power limit will therefore also depend on the impedance of the load.