Does the "timing belt" on a 2002 Prius also serve the water pump?

The service docs don't really recommend that procedure ... if the charge is too low to make the low pressure switch and allow the compressor to run, all they do is with the engine off connect to the high side service port with the engine off, and with the engine off allow enough refrigerant into the high side with the engine off to create enough pressure (over 33 psi is enough) with the engine off to close the pressure switch. Also, all that should only be done with the engine off. Then be sure to close off and isolate the high side from the refrigerant container, all with the engine still off.

Then after making sure there is no way the container is connected to the high side, you can run the engine and finish charging through the low side as you'd expect.

They do suggest, if the charge is that low to start with, there might be a good chance of having air in the system too, which means just adding refrigerant won't give a satisfactory outcome unless the air is vacuumed out first.

-Chap