Wow, this is definitely a very unique situation!

First things first, if you haven't already talked to your vet about your dog's excessive thirst, you need to get him checked out and have bloodwork done. Excessive thirst (to the point where the dog is bloated and lethargic after drinking) isn't normal and could be a sign of an underlying health condition. Off the top of my head I'm thinking Cushing's disease, liver problems, kidney problems, and diabetes. It's definitely important to get that checked out.

My suggestion would be to get one of those indoor dog toilets -- there are many brands to choose from, but they are usually a raised platform of artificial turf that collects the urine in a tray. I'd put the dog toilet outdoors in a sheltered place where it can stay dry (such as a porch or patio) and take him there when it's time to urinate. You can easily hose it off to clean it once a week, since it's outdoors already. Another option would be to create a sheltered corner of the yard where he can go to do his business, using either a tarp or slanted roof to keep the area dry. Gravel would also be a good idea, as it should drain well.

Finally, get your housetraining game on point. Make sure he's either in his crate (when you are at home) or in a long-term confinement area (when you're away for more than an hour.) The LTC area will have a bed at one end, a chewy toy to play with, and a toilet area at the opposite end from the bed. When you're home, every hour, you take him out of his crate, put him on leash, and then jog to the toilet area, where you say "Go pee!" When he finishes peeing, give him three treats and praise him. Jogging him will jostle his bladder and he'll be more likely to pee when he gets to the toilet area.

If you follow this routine meticulously for two or three weeks with no accidents, you will have a completely housetrained dog that will pee on command. Assuming he's healthy and just likes drinking from rain puddles, you can then let him drink from a puddle for a few seconds as a reward for peeing. So the way this would work is: "Go pee!" > Dog pees > "Good dog! Go drink the puddle!" > Jog him over to the puddle > Dog drinks from rain puddle > Praise. If you give him the puddle drinking as a reward, he'll be the fastest urinator on the planet because he's going to want to pee quickly so he can go drink from the puddle.