The grounding is accomplished through the metal pipe, not through the water.
Pure water is an effective electrical insulator. It's only conductive when there are dissolved salts in it (not just NaCl, but any metal salts creating free ions floating around in the water). In that case, water can become a very good conductor. But it isn't really the water doing the conducting.
That PVC pipe is not creating a suitable path to ground.
You need to drive at least one grounding rod (probably an 8' copper or galvanized steel solid rod, which you can buy at your local home improvement warehouse), and run a #6 copper wire from that rod to your main service panel, where you'll bond it to the grounding bus. You also need to leave all of your copper and/or iron pipes bonded to the grounding bus in your main panel.
Most lamps have a two-prong plug. All the lamps in my house, even the ones with metal casings, have two-prong plugs. I don't think you have a safety problem.
However -- if I were building this lamp, I would attach a three-prong grounding plug and cord, just because the aesthetic of the lamp is so obviously 'heavy industry' and most heavy industrial equipment is fault-grounded.
In order to do this, you would need a grounding lamp kit. It's better and cheaper to purchase the whole kit, but if none is available, you can safely substitute a three-prong grounding plug and cord into the kit shown in the Home Depot tutorial.
You will also need a grounding locknut, also known as a bonding locknut. These are widely available if you know what to ask for.
I think the design calls for a hole through the pipe, but the rest of the project requires no cutting or drilling, and the hole is not strictly necessary.
Hole option: before beginning assembly, drill a 3/16" hole into the side of the 3/4"-to-1" coupling, through the 1" section. Drill about 1/4" from the end, or just clear of the reinforcing flange. Chamfer the ends of the hole because you will be feeding a wire through it later.
In step 9, you thread a 1/2" grounding locknut onto the 1/2" close nipple, or thread a 3/4" grounding locknut onto the 1/2"-to-3/4" coupling. Snug it up tight. The threaded rod from the lamp kit may have a hole for the ground wire. If so, pull the wire out through the hole now.
In step 12, if using the hole option, feed the fault-ground wire from the grounding lamp kit out through the hole. If not using the hole option, bring the wire out through the open end of the 3/4"-to-1" coupling. Strip the end of the fault-ground wire and attach it to the screw on the grounding locknut.
The pipe assembly is now safely connected to the round pin on the three-prong plug.
Best Answer
You should be able to purchase a three wire cord with a molded three prong grounding plug on one end open wire ends on the opposite end. These are often sold as electrical appliance replacement cords and should be available at most good hardware stores and big box type outlets.
Here is an example of one type of these cords:
Make sure to use proper strain relief on the core end at the lamp to keep stress off the internal electrical connections.
These cords are very similar to if you got a three prong extension cord and cut off one end. You could do this except in many cases any safety agency listing for that extension core becomes non valid once the cord is modified. This may also be a reason to check if the appliance replacement cord that you purchase is marked with a safety agency listing/logo.