I had written up this question and was experimenting before I posted it in anticipation of what I thought you all might suggest I try. I'm pleased that that experimentation led to a solution. I'll post it just in case someone else has the same problem.
On the Tablet PC Settings applet there is a Setup button. When you click it the internal screen changes to a "touch this screen or hit enter if this is not a touchscreen" message. I had tried this once before but since the touches went to the external monitor I wasn't able to complete it. This time I used the mouse to click on the form when it showed on the internal monitor. Then when it showed on the external monitor I hit enter to indicate it isn't a touchscreen. That fixed the issue.
I have no idea how the laptop got into that weird mode to begin with since I'm very certain I've never run that Tablet PC Settings applet before.
Now if I can just figure out why the touchscreen generates phantom touches under Windows 8...