Thank you, all who contributed to this. You've just made things a lot easier for me. Some of my clients didn't know how to right-click a file and what to choose to save the file to their computer. Even with the instructions right there on the page, it was too difficult. Now when they click on a link, their browser asks whether they want to run or save the file. Excellent!
I did make one modification: I put direct_download.php in a dedicated folder and link to it that way: <a href="[my catch-all folder]\direct_download.php?file=..."> That way, I don't have to remember to copy the php file to every folder in which I'm storing files for download, and I can still use it on any page on the site.
I need to make my site more accessible, especially to mobile users, and more standards-compliant, so I'll be studying here when I have the time.
-- Timothy J. McGowan