Testing?....
To test the charging circuit on the ski, you don't have to hook up the water to it. But, you have maybe a minute to run before you'll go hot.
When you changed the rectifier, did the battery start the motor? If so, then your battery was likely charged good enough to run. The reading you posted above with the ski running, shows the charging circuit is not working.
When you replaced the rectifier, did you replace with the proper one. They are all not created equally, you have to have the correct rectifier for your motor.
Check all your fuses. If your rectifier is the correct one, you may have a bad fuse or you've shorted it out, probably not knowing.
Never work on your electrical system with the battery connected.
You turn the water on after the engine starts for the reasons timmyboy states, but also, because the pressure from a water hose can breach the "0" rings that seperate the combustion chambers from the water jacket. :cheers: