My cat understands pulling to open a door.
Counterintuitively, he does not understand pushing to open a door. Even if the door is somewhat ajar already, and all he would have to do is apply the lightest pressure to swing it forward.
Rather, he will stick his paw around the side (or even bottom) of the door, and pull it toward him. Which will shut the door he wants to go through. At which point he will meow piteously until I let him through the door.
But even more oddly, sometimes he will use the OTHER paw, and try to open the door by pulling the DOOR JAMB toward him.
This method actually sometimes gains him entry, because the act of pulling the jamb sometimes pulls his body forward, which then will nudge the door open enough for him to get his head through, and the rest then follows.