I consulted Wikipedia about the design of german light water reactors. Bor is used in addition to water. It enables regulation of the power (in some limits). Makes sense to do that. I don't know if this is done in Japan too. It could explain, why the reaction does not stop after the water is gone.
Where did you get that assumption from? Water is definitely the only moderator in this form of LWR (based on the published design specs).