It isn't an actual war. Humans were flying to the moon, planting flags and acting as if they "owned" the moon. Furthermore they're planning on building a moon base, and other such things. The Lunarians saw this as an act against them and decided to "go to war". Really, humanity has no idea that the Lunarians are on the moon, because of the Lunarians' barrier. The Lunarians, likewise, are unaware that the humans are unaware of their existence, and treat the lunar landings and expeditions as invasions. Nothing suggests that there was actually a battle taken place on any of the Lunar wars, despite the rabbits' claims otherwise.
The rabbits' actions, instead, were likely to tear down the flag that was left by Apollo 11. That was their "warfare". After this, Reisen escaped the Capital and came to Gensokyo. Presumably, she brought the flag, since it's currently in Kaguya's possession. The funny part is that the Lunar Capital didn't know what caused the flag to disappear, and that among other odd events caused the Capital to become suspicious for a possible traitor. Their first suspicions were the Watatsuki sisters, and this unfolding of events is what caused the second "Lunar War" that starts the events of Imperishable Night. The suspicion of the Watatsuki sisters is also part of why they later came to Gensokyo in the first place.