Since I couldn't find details about how this particular item goes together and I have the resources to, I went ahead and bought a new kit so I could tell how the pump and tank attached to each other and so tell how to disassemble the existing one. (Unfortunately my first purchase was of the plastic-tank model, so that's what the pictures are of.)
Sure, there might be other ways to do that, but that's what I did and now you don't have to. :)
Hopefully what I found and learned will augment the existing how-to videos to make someone's life easier.
The packaging is simple and robust. It's a cardboard box, a styrofoam insert, and a couple of papers:
You can see in that last photo the tank and the pump assembly.
The tank in this particular kit is just one piece of plastic. The neck has threading on the outside.
The pump assembly slides apart into two sections that I'll call the pump and the mount. Here's the pump:
This is a single unit, already attached and glued together. Normal operation has this sitting snugly in the mount and lifts out as a unit to allow refills. The problem I had is that the translucent straw on the pump broke off and fell into the tank.
Here's the mount:
Note the brass nut and the rubber gasket. These function just as described in numerous online tutorials to seal the sink surface off from the underside.
The part that I could find no photos of anywhere is this:
The bottom of the mount has threading on both that outside for the brass nut and the inside for the tank. Now I know I need to retighten the brass nut and remove the tank before removing the nut and mount. Silly as it may seem, it's now on the Internet.
Thanks for reading.