The next day I tested the pump by pouring water in the base and running it for a couple minutes. It was really exciting to actually see liquid coming out from the lines, although it was pretty sloppy looking. So I had a good pump, the vessels were clean, and it was time to fix the broken lines. What really helped me here was the fact that the manufacturer's design and components were basically idiot proof. The grommets (eyelets) and string seem complicated and intimidating, because if you mess with one out of curiosity, you could endanger the whole project if you break it. But, mine were broken already, so I went in with flathead screwdriver and some needlenose pliers, ripped out the grommet, threaded fishing line through the bottom and pulled tightly back through the top and voilą!
Now I was ready for the first test. I decided to hang the lamp and then fill her up, just to prevent any accidents. She took 7 pints of oil (I actually had to run back to the store to get more oil). It definitely was a nail biter waiting for the oil to drip out the first hole. But after about 10 minutes, I was rewarded and the drips quickly moved around the circle until finally it was complete. Her name is Venus :).
The lamp has a certification sticker from the city of Los Angeles that reads "AL 699563", but no manufacturer name or date or serial number. Since Los Angeles was the home of Johnson Industries, possibly this is a Johnson Industries lamp, but TJ conjectured it might be a copy made in Mexico by some unknown manufacturer, which seems more likely to me, too. The lamp's ivory finish and ivory colored statue are characteristic of Creators, Inc. rain lamps, but Creators rain lamps tend to be stamped with their name and even exact dates of manufacture, and their statues I've seen are different than this one, so this is probably not a Creators rain lamp, despite appearances.

Also, when TJ obtained this rain lamp, both the lid and bottom basin were bolted to the central cage part with a very strong, clear glue.