The "same bad behavior from other external" does make "computer problem" seem to be the primary culprit. I wonder if the newest external can be returned to a local store, or if it was an on-line purchase?
I might try to get a 3rd unit from a local store that has some tech-department and, just in case it too exhibits the same behavior, see if the techs can find a problem in a Windows setting (or 2 or 8 - the joys of the registry).
But, before that...
I wonder what Auto-Play settings are for that drive, and do those follow along with any changed USB port? (Your computer probably has at least 2 USBs... does AutoPlay work on one but not on another? Therefore, AutoPlay would be enabled by the USB port's address, and not the External Drive's driver.)