HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

by the way, the message box asking about a quick battle only appears for the blue player. The red player doesn't have it. Instead, they immediately see the battle results window + the option to replay.

This shouldn't happen; it needs to be fixed.
The question about a quick battle should only appear if the game is running without a replayable quick battle! (i.e., in HD+ mode, the corresponding option is disabled before the host starts the game, or the game is running without HD+).
When a replayable quick battle is enabled, the question about a quick battle (which is needed as a safeguard against a forgotten unchecked box) is not necessary because the result of the quick battle can always be replayed manually.
Well, it's not a critical bug, so we don't need to rush to fix it.