Thanks to a hint from a user on another web forum, I have found a fix, at least for now.
Specifically, a normal "master rest" on the eTrex Vista is accomplished by holding down the "Page" button and "Enter" (i.e. the click stick) while powering up. You are prompted that a full reset will occur, and after confirming, the machine is reset. However, in my case, this achieved a full reset, but did not fix the date problem.
However, the undocumented "super master reset" on the eTrex Vista is achieved by simultaneously holding down the "Page, "Enter" and "Find" buttons while powering up. You have to hold all buttons in for about 5 to 7 seconds, and you DON'T get the warning message, but when the machine finally restarts, it is fully reset. After acquiring a satellite fix (took several minutes) - lo and behold - my time AND date both seem to be correct!
Oddly, neither Garmin tech support nor the local reseller seemed to know about this method of REALLY resetting the machine - I will pass on the information to them, for future reference.
The link which documents the "super master reset" procedure is here: http://www.gpsinformation.org/dale/secret.htm#reset
Hope this might also help other eTrex Vista owners who are experiencing the same date problem.