Speaking of which, we pulled the plug on the UPS to make sure the auto-shutdown software worked. The UPS began to bleat, and the desktop computer announced that it would shut down in a few minutes. After a few minutes, it did indeed shut down. Meanwhile, our laptop computers could still reach the internet. So that's all working, and feeling better about things here. There's only one potential hitch, and that is that the desktop machine will shut down, but will not turn back on when the power comes back up. That could get annoying if we're both away for an extended period of time. (I have the same situation at work; a few days ago, I had to have a co-worker go turn one of my computers back on because we had a power failure and that machine doesn't automatically come back up if the power fails.)
1. When the UPS kicked in, the "Site wiring fault" light lit up on the power strip (manufactured by the same company that built the UPS, APC).
2. A few seconds after we we plugged the UPS back in, the power glitched briefly, causing the DSL modem to recycle and my computer on its docking station to announce that power had failed and it was disabling the dock. After the glitch, the "site wiring fault" light turned off.