Example 2 is a famous spot in the finale of that old warhorse, Beethoven's Fifth. This is near the end of both the fourth movement and the coda of that movement. The question is whether the timpanist should play the ink, which follows the lower strings and contrabassoon, OR reverse the quarter note and rest and thereby follow the rhythm in all the other wind instruments, including both horns and trumpets.
The common correction made here in red that reverses the note and rest assumes that the printed score is incorrect. For such an error to persist in what is no doubt the most popular symphony ever written makes one wonder if it is indeed a mistake. Having tried both versions myself many times, the "ink" verison just seems wrong in context as inappropriate for Beethoven. Indeed, if one looks at the writing both throughout Beethoven's music and in the remainder of this work, the timpani part is often in lockstep with the trumpets. Here all of a sudden for this one brief moment the timpani abandon their "dance partners" and the orchestra "bounces off" the timpani note instead.
Another way to study these questions would be to take a musicological approach and consult all the available evidence in all possible sources, including early drafts and sketches. For these examples I have not gone to this level of study, preferring Rule #3: