Either the documentation is wrong or the implementation is wrong (a bug).
I understand the actual behavior, now. My initial question had to do with understanding what the documentation meant. The responses I've received have ignored the documentation of the PopupMenu option and only described actual behavior.
The description "display like a popup menu with initial value name" says this looks and acts like a PopupMenu, but it doesn't. In PopupMenu, the first argument, name, does set the initial value for the menu. I don't have to manually make a selection—what I see is what I get. With ActionMenu and the Appearance->"PopupMenu" option, I get a menu that looks exactly like an ordinary PopuMenu, but does not behave the same. It seems that either the documentation author erroneously assumed equivalent behavior, or that equivalent behavior was intended as documented, and the actual behavior is a bug. At this point I would declare it a bug, since if it looks like a PopupMenu it ought to act like one. If Wolfram doesn't consider it a bug, I would call it a design defect, and the ActionMenu documentation should clarify how the PopupMenu option differs from an ordinary PopupMenu.
My pursuit of this might seem nit-picky, but put yourself in my place. You asked what behavior I need. The answer is, "I don't know yet." I'm developing an application for which I want a UI that doesn't require typing expressions into a notebook. I'm so far very pleased with what I'm learning in this study group, but my purpose here is to try to understand the whole tool set that is available to me. When the implementation doesn't behave as described in the documentation, I waste my precious time trying to figure why I can't make it work, and I waste your precious time trying to help me.
