Does the game have a specific time it turns villages of a player who has been inactive two weeks into barbs? Like I understand 14 days, but like 00:00, 09:00, etc.?
Nah, though it can take much longer than 2 weeks if the player has ticked the "Extend automatically" square for pp. THen the account will be alive as long as there are pp-points awailable and not turn grey untill 2 weeks after they are gone.
As far as the original question of what time on the clock will they turn barbarian, perhaps it is 14 days from the exact moment in time (e.g. 23:47) that they either last logged activity or that their PP ran out.