It turns out that Exit Application does indeed have an effect, it's just a little more subtle that I was expecting, and still not the effect that I was hoping for (an application logout). When you instantiate a new window in WebDirect, they "stack" up in the current browser tab. If you're simply executing the New Window script step of button action, you'll see a number appended to your browser tab name that increments each time you execute the New Window action. This behavior seems very similar to the behavior in FileMaker Pro except that you're not getting an actual new window. If you execute a New Window action a 7 times, you'll have a tab with a name like "FileName - 8". If you then issue a Close Window action, the tab number will decrement for each action. When you close the last "Window" you will be logged out of your application.
It turns out that if you have a file with an OnLastWindowClose script configured, the Exit Application action will close all windows but the last one. So any intended logout action could (should?) be scripted as either:
Exit Application Close Window [Current Window]
-- or --
Loop Close Window [Current Window] End Loop