When is a courtesy runner allowed to join the game according to the rules presented?

A courtesy runner is allowed to join the game specifically when a player who has been the catcher in the previous inning is on base. This rule is in place to ensure that the catcher is sufficiently rested and can perform well in their defensive role after having been active in the previous inning. By allowing a substitute to run for them, the game maintains its pace and the catcher can prepare for their next turn at bat without undue fatigue.

The other scenarios presented are not valid under the rules governing courtesy runners. For example, limiting the use of courtesy runners to only after the second inning or when there are no outs in the inning does not align with the established guidelines. Similarly, allowing a courtesy runner when a player from the defensive team wishes to rest does not comply, as the courtesy runner rule specifically pertains to catchers who have batted in the previous inning. Thus, the correct answer highlights an important strategic aspect of the game that provides relief for critical positions like catcher, ensuring they remain effective throughout the contest.