Okay, so I solved my own problem. :-) I was looking at Keynote preferences under the "Presenter Display" section, but I never looked at the "Slideshow" section. Turns out the settings got changed around on the "Present using Primary Display" and "Secondary Display" setting. Once I got them fiddled around correctly, it worked.
I have to say that having those settings in one section and the "Use alternate display to view presenter information" in another section makes it easy to overlook.