After a bit of work, I found the answer.
Here is a letter I sent to the original blog author:
I ran a test on a spare HP laptop I have. I can give you the gory details if you want, but the bottom line, is that on my test computer with Windows XP SP3 and Office 2003 Pro SP3, the timer didn’t work on my test laptop the way you describe.
In my test environment, the Word document “Total Editing Time” timer will run under these conditions:- The Word document needs to be open.- There cannot be another Word document open as the primary document.
With a new MS Word document, the timer continued regardless of if I opened a web browser and surfed for a while, regardless if I minimized the Word document, regardless if I opened up an Excel spreadsheet and worked on that. The only way I could stop the MS Word timer was to create another Word document, and have that new Word document be the primary document.
If you want to see my hand-written notes on this (archaic, eh?), see the gory details here.