"That colony might say that since resources on the Moon are so precious, water so rare, there is not enough for another colony. Any additional colony is really an invasion of their colony. In effect, the Moon belongs to them. Wouldn't matter how true it could be. After all, the colony would grow. Could say that."

Bowser I'm impressed. This is a good point.
If there is say 6 billion tons of water at the lunar poles, this isn't really in practicial terms a scarity, but it could easily be argued that it was.
Additionally, the Chinese could effectively cut Earth into two powers by owning the Moon. They could "own" both poles and they could afford to "give up" one pole for rest of the world to "have".