To confirm my problem and to solve it, Garmin International yesterday released 35x7 and 2xx7 update 3.80 where they said among other fixes:
Fixed issue with device offering to update certain files after they were previously installed.
I hope that that ends this thread.
In order to get the update you may:
[Only registered and activated users can see links. ]
or to use WebUpdater.
As my unit is in Garmin hands let's see did and how they will solve my problem.
Even fix is not confirmed yet, I think thread can be marked as solved and to be closed.