Actually, water is used in the light-dependent portion of photosynthesis, not the Calvin cycle.
The added water efficiency doesn't actually relate to water's involvement in chemical processes of photosynthesis at all. Water is instead lost incidentally through stomata when they're opened to allow CO2 into the leaves (see here for explanation). This CO2 IS required for the light-independent Calvin cycle.
Both C4 and CAM plants preserve water by decoupling (in space or time) C-fixation from the rest of the Calvin cycle (see here). They instead "store" carbon in 4-carbon molecules to use as a C source in the absence of CO2 (i.e., when stomata are closed). In doing so, they can perform the Calvin cycle without receiving more CO2. As a result, C4 and CAM plants can photosynthesize without having their stomata open all the time (or during the heat of days for CAM plants). The final result: less water lost through evaporation out of the stomata