Device can be powered at 6V or 12V, which is better? Why?
I suspect there's an internal voltage regulator to regulate it to 5-6 volts. For compatibility, they added this voltage regulator so you can also power it with 6-14V. Internally, it's regulated to a lower voltage.

That would mean the voltage regulator dissipates half of the power in heat when you use 12V, and heat is bad (not only because it's a waste of energy, but also because it might harm other components). Therefore, a lower voltage would be better.

Disclaimer: I only suspect there's a voltage regulator inside, I don't actually know it.