Yes, the main coolant intake line is the large black line connecting to the pump right above where the drive shaft goes through the hull. The jet pump provides the pressure to push water into the motor and then out the exhaust, and a couple other smaller lines. The faster the pump spins the more water is pushed through the motor.
My main ride is a 94 XP that I rebuilt the motor and pump, crank, pistons, pump shaft and bearings impeller, liner, carbs.... It was trashed when given to me.
If you rebuild the motor rebuild the carbs with original Mikuni carb parts. Any motor rebuild should include the carbs since they are the root of many motor failures.
Replace the grey fuel lines with new fuel lines, the grey ones deteriorate inside and cause carb/running problems.
Clear out all coolant lines and passages in the cylinders, exhaust manifold and pipe.
Just to clarify, the piston looks like it got hot? Or were you refering to the cylinder head showing discolored paint as a sign of overheating?
BTW - Welcome to the forum!