One recommendation is to update the system's BIOS to at least revision A07 (or, preferably, to the latest revision A11). Check the setting for Wake On LAN again within the BIOS after performing the update, and disable it again. Save the changes, reboot the system, and shut it down.

If the system still boots whenever the Dimension 8300 is powered on, I would recommend instead connecting each system to a hub/switch/router device via a standard patch cable, and networking the two systems in that fashion.

Thanks for that, The links you gave were for another system but I went on the Euro Dell site and downloaded A08, which was the latest version for my system, and so far, the XPS isn't booting up when the 8300 starts up.  Once again, many thanks.