I followed several lines of advice at that link. Something worked. I'm not sure what, but here is a summary for those who may find themselves in a similar situation:
Update - As strange as it seems, I have now noticed that the little systray slider/arrow has returned, apparently spontaneously! As I noted previously, I did install the
Unplug n Pray disabler, but saw no immediate results. I also installed the Windows XP
Tweak UI power toy from Microsoft. In Tweak UI, you may want to try clicking "Taskbar and Start Menu," and then select "Grouping." When you do that, you will notice a topic to your right entitled "Button Grouping." Under that heading, you will see three radio buttons that you can select from to alter grouping behavior. The default is "Group least used applications first." Before you make any changes to the selection, note that the "Apply" button at the bottom is grayed out. To re-enable the "Apply" button, select either of the other two options (it doesn't matter which), and then reselect "Group least used applications first." Notice now that the "Apply" button has been re-enabled. Click "Apply," and then click "OK." I did that and didn't notice anything immediately, but whether because of doing that or because of installing and using the Unplug 'n Pray disabler, I do now notice that the slider/arrow has returned and my systray icons hide as they should when not in use. Another Windows Mystery!