The Moonlight Stones appear only with acts of true love.
One appeared to Masami and Nozomi when Masami saved Nozomi from drowning, one appeared as Himeko and Tomoki bonded on the roof, and another one appeared after the new Shachihoko are delivered to the palace. The Moonlight Stones appear when someone feels any really strong positive emotion (Happiness, relief, love, etc...)
Masami got hers when she realized that Nozomi was ok. Himeko got hers when talking about her dream to create shachihoko. They come down as a rather common occurrence in Kyoto, and there's near constant happy rock music playing there.