"But a long term sum of results is a completely different game."
That's exactly the point! A naive intuition about dice is that they have a single probabilistic long-term score. E.g. 6-sided die is 3.5. So if one dice beats another, it should be transitive (the intuition goes). It's the summation (implied by averaging over time) that leads to the naive intuition.
Maybe this description of the two games might help clarify the link.
Game 1: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b) dollars.
Game 2: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b > 0 ? 1 : -1) dollars.
You can kind of squint and see that game 2 is the same as game 1 just with (a - b) "rounded" to either 1 or -1.
I also don't understand how normal dice play counts as "premature rounding". It's just how playing dice works -- you compare the numbers versus each other.