When the "8400" ended up with major hardware problems, I purchased a used "4700" which did not come with a OS installation disk so I am using the XP installation disk from the old "8400" and still legal.
The answer in both cases was to ensure auto-update/auto install was turned on, then leave the PC online for a few hours.
I have disks for SP1, SP2 and SP3 which I purchased directly from Microsoft.
Both SP2 and SP3 were installed on both computers as part of the XP installations. IMO, you should have attempted it as soon as the install with the primary XP disc was complete. But since you mentioned that "SP2 and SP3 were installed on both computers as part of the XP installations" it seems that you may have waited until after you installed the SP's. forgive if I missed something, or got confused somewhere...1.
For this reason, I had to go ahead and install SP2 and SP3 in order to access the internet and perform the activation.1.
I have not used the diagnostic tool at the web site linke which you included in your message. Since there are similar, but somewhat different problems which have occurred on my friend's Compaq and my Dell, To keep them separate, my friend's XP disk for his Compaq is the "Home" version and mine, for my Dell, is the "Professional" version.