Not sure why, but clearing the market cache and data seemed to fix the problem. I'll try to write up my experience for posterity...
Before clearing the cache, the only categories that I had in "My Apps" (i.e. going to the Market app, hitting the menu button (three vertical dots) and selecting "My Apps") were "Installed" and "Not Installed". (By categories, I mean the apps listed under a gray title bar displaying the category name.) After clearing the market cache (as described by Tidbits in this thread) and going back into "My Apps", I had the category "Manual Updates", under which several apps were listed. By selecting an app under this category, I would be brought to the app page where I could manually update it.
It seems odd that the installed apps are listed in two different places; in the settings menu (as described in this thread) and in the Market app itself. In the settings menu you can force stop, uninstall, clear data and clear cache, but as far as I can tell, you cannot update the app from the settings menu, or tell whether or not there is an update. Instead, you must go into the Market app (which is an app itself) to do that. Just kind of seems redundant or non-intuitive to this newbee.
Anyway, thanks to all who helped me to work through this.
