The water itself would have some conductive properties but they should be very low as specified by AWS J1.2:2016. The machine, transformer and control should all be grounded. Therefore, the water does have a potential path to ground.
WELDER SCHEMATIC WITH COPPER CONDUCTORS
In the resistance welding process the secondary voltage is low, frequently 4-6 volts. With this low voltage the current is going to flow through the most conductive path. This is the copper and copper alloy conductors present in the form of conductors, shunts, cables and electrodes. The water potentially can conduct but is highly resistive relative to the copper conductors and very little conduction will be present. The water conduction will be minutely small in comparison to the thousands of amperes conducted through the copper system. This minimal flow pattern in the water is the same in AC or DC systems.
Reference: AWS J1.2M:2016, Section 4.5
Resistance Welding Manual, 4th Edition