First the falling flat:
As you can see from the top two plots, the highest and second highest scores tended to come in the first inning, not later in the game. In fact, there's a strong drop at the end, suggesting that relief pitchers do help keep the score down.
I conclude that the high score doesn't come when the pitcher gets tired. I was wrong. In fact, it looks more like the pitcher isn't quite in the game yet, or the team hasn't gotten synchronized for play yet.
Now let's look at the second-highest score distributions, to see what we can see.
The distributions of highest and second-highest are roughly similar, except that the second-highest inning's score is less than half that of the highest inning's score, on the average. And the reduced score distribution for 7 innings (the extra inning games are a small effect here and I ignore the correction) looks much more Poisonnian, dropping by about a factor of 2 each time. That's on the lower right: compare with the 8-inning reduced score on the lower left.
I used the 7-inning reduced score distribution to predict the highest and second-highest innings' scores, and those are in the bottom row of the figure below. Recall that when you are looking at the highest score of a list, 0 is not as likely as higher values. And they have the same general shape as the real distributions above, but with much smaller averages. The average real highest inning's score is 2.3 but the estimate's is 0.7, and the average real second highest inning's score is 1.1 but the estimate's is 0.5.
What can I conclude from this?
The highest inning's score is in substantial disagreement with the estimate. So is the second-highest inning's score, though the real distribution could be a combination of the background estimate and something else.