Add on domain does not have access to "../privatefiles" at public_html level
OK, I figured out what was going on and how to fix it.

The problem was indeed with the "../privatefiles" declaration for the private files directory. Just like for the old DOS files, the "../" means "one level up" and that is okay when the core file are in the www directory.

But when they are below or way below, the "../privatefiles" has to be replaced with the full path: "/home/whatever/privatefiles"

Now it works well.