Thanks for the offer of help. To have Windows Explorer display the Date Picture Taken, switch to Details view, then right-click on Name (or any other column heading), then select More down at the bottom and then check Date Picture Taken.

But I think the problem I am having actually has nothing to do with the Exif information or Exiftool. The problem is that Flickr Uploadr doesn't seem to use "Date Picture Taken", even though it says it uses that field. Instead, it uses the Windows field "Date Modified", which is different.

Sigh! It doesn't appear I can change the Windows field "Date Modified".