Bragging Mechanic

Bragging will easily translate into Points in most players minds...

One thing I wasn't clear on by reading the rules was what makes a "better" brag.
The example helps, but still leaves it unclear.
- 2d8 to 2d6 reduces the dice size, which is obvious enough.
- 2d6 to 1d12 reduces dice count but increases dice size, which causes my confusion.

Could Coop have instead bragged 1d20, which still reduces dice count but increases dice size?
Likewise, technically by the rules Dash could have bragged 4d10, because it reduces dice size.

Munchkinism aside, as I understand it the intent of the "better brag" rule is more like: "Each brag must be better (that is, lower) than the previous. Multiply the dice by the sides (so 2d6=12 is better than 2d8=16). On a tie, the lower number of dice is better (so 1d12=12 is better than 2d6=12)."
Is that right?

As an aside, I agree with the previous poster that "boast" fits better than "brag".
(Further aside, it immediately reminded me of the Clans in Battletech.)

Regards,
kos