I return to the language question.
If the Captain thought he had told the P2 to push down, then he would therefore not say 'I have control' because he thought he was pushing in the same direction as the P2. Likewise the P2 would also think, thanks to his interpreting the "pull down" command as "pull" (which he was instructed to do later), that the Captain was either not touching the controls or trying to move the stick in the same direction as himself.
The differing misinterpretations of the Captain's "Pull down" command explains the subsequent actions of both.
But that doesn't explain all the actions of the captain. If he thought the FO was doing the right thing, the captain wouldn't have touched his own sidestick. He would have let the FO fly the plane.
The captain's problem was in trying to take control without telling the FO to hand over control.
The three words "I have control" might well have saved this aircraft.