Then, after removing the foliage and making sure the power cord was disconnected, I slid the two decorative bands out of place, as shown below. The upper band slid upwards and the lower band slid downwards. The lower band cannot be slid past the bottom of the basin, as far as I can tell, although the top band obviously can be slid off the top completely.


Both bands are made of a rubbery material and did not appear to be fastened with glue, screws, or any other means other than friction. If these bands are to be left off the lamp for a few days, it's better if they are stored flat, since standing them on edge for a long time will cause their circular shape to become distorted.
Some types of rain lamps do not have decorative bands at all, so the screws are already visible, as in this unknown brand:

The only practical way to remove the bulb is to reach in through the strands to unscrew the bulb, and then if desired, to pull the bulb out between the strands. To avoid getting oil on your hands and wrists, first use a paper towel to wipe down the strands that will come into contact with your skin. While pinching each applicable strand with a paper towel, run the paper towel down each such strand.
Tip: To avoid accidentally dropping the light bulb inside the cage, I've found it very useful to hold the bulb with two hands by reaching in from opposite directions of the cage at the same time.

With the decorative bands removed, screw heads were visible, three screws at the top, and three at the bottom. These required a flathead screwdriver to remove. There was a line of dried glue that ran across all the screw heads and encircled the cylindrical housing holding the screws, probably to discourage disassembly, and/or to void the (35-year-old) warranty if tampered with.
TJ reports that the screws on his vintage rain lamp were sealed with metal epoxy, and therefore took a great deal of force to unscrew. Since my screws were also covered with a type of glue, it would seem that rain lamp manufacturers were attempting to discourage the removal of the bottom basin, which is puzzling since rain lamps can never be cleaned well unless the sludge from the bottom basin is exposed for direct access.
Unlike the bottom section, the top section of the lamp would not completely and readily disconnect from the cage part. The reason for this is that the light bulb is screwed into the lid, and protrudes through a hole in the metal that is in the cage part, a hole that is smaller than the width of the bulb at its widest point. Therefore the top could not be removed further (without breaking the bulb) unless the bulb were removed first.



The little cap nut on the very top unscrews, on both styles. On the table lamp, unscrew it and then lift off the top. On the hanging model, it has wires through it, so the top will only come off and to the side, it won't completely separate.
As with the canopy screws, there may be a line of glue over the screw heads on the basin. A flathead screwdriver was all I needed to remove all exterior screws on my lamp.

Once the screws were removed, the cage part of the rain lamp could be removed completely from the bottom oil basin.



Obviously the brass collecting basin gets filthy after not cleaning for a few decades! I also found a stem from one of the plastic plants at the bottom of the basin.
The following is how the electrical wires leading to the bulb on my lamp are connected. I suspect other rain lamps are very similar, with only the wire colors changed.
The original electrical tape was black, and I also used black for its replacement. To help prevent a repeat of this accident that tore the wires apart, I wrapped a piece of duct tape around the group of four wires coming from the top of the electrical post, so that if the lid falls again, this extra piece of tape will hopefully take the stress instead of the junctions of the wires. I believe the rain lamp should have been designed with this extra protection from the start.


The printing along one of the wires reads: GENERAL ELECTRIC.
http://www.faqs.org/docs/electric/Exper/EXP_4.html (12-16-05)



The following photo by Connie shows such a shield-mounted threaded post for the statue:
