The rule that you are looking for is 804.05 A(4)
The other rule in question I think can be found in the competition manual and talks about the time frame of 'from the 2-minute warning to the time that the cards are turned in'.
My interpretation of the rule is: as long as the cards have been added and all players agree on their scores (initialing the card is optional), if one player offers to take the cards back to Tournament Central that is the point that the round is over for the rest of the card. That one player can assume the burden of 'extra time' for his card and his round ends when he turns in the cards. But that's just how I roll.