Well, I'm a senior electrical engineer working for a semiconductor company, so I'm pretty much an authority on the subject.

If you don't believe that LEDs are insensitive to changes in voltage, I invite you to pick up any first-semester electronics textbook. A forward-biased red LED has a potential drop of about 1.2V across it, no matter how much current you're providing it. The provided current, of course, will change the amount of light output.

The light output difference between your 6V and 12V batteries probably has little to do with the LEDs themselves, but is probably due to the different equivalent series resistance of the two different battery packs. Smaller batteries have smaller plates, and thus have larger equivalent series resistance and lower current maximums. I'd have to take a look at the specific design of your light, but it's unlikely that it's been engineered to accept a range of 6-12V without using either some kind of voltage regulator, or a large number of LEDs in series. In either case, it's the higher current supply that makes the light brighter with the 12V pack, not the increase in voltage over each diode.

- Warren