I have the Kickstarter version from Vainglorious Games and in that version card powers only activate on the round they are played.
So if you played a 9 that stayed, and the next round an opponent plays a 9, both cards would stay. (Unless another 9 was played at the same time as your opponent's. Then both would go away but yours would stay.)
Then it becomes a question of round because it the rules it mentions that the round ends after the 6th card is revealed, but the description indicates that a round is when cards are revealed. What you are saying makes mores sense. I guess we misunderstood what a round is.