Fairly sure "juice" means "power" - as in P= V*I.
Regarding the HE-6:
InnerFidelity measured them at 20 mW/90 dB ->1.25 mW/78 dB. Too lazy to take that quarter off. Makes me want to agree with 83.5 being optimistic.
Anywho, I agree with:
"I just feel its a manifestation of this desire to have your device perform better, something that is actually governed by the limitations of the product design." and would add:
In some cases a person gets an item, it's incompatible with or performs poorly with their existing items, so they over-buy to compensate, and feel that over-purchasing was the "only answer" to the problem. I've seen this pan out a time or two in context of "juice" (you have a headphone that's relatively or blatantly inefficient, like the K701, HE-6, K1000, etc and someone tries to run it with something silly and it flops, so they run out and buy something HUGE in response to positive reviews/hype, and then champion that HUGE purchase as the only viable solution). While it doesn't result in bad advice, per se, it can build up a dragon when one doesn't need to exist.