The ball's location is the only thing that matters.
A player can stand outside the field and play the ball legally, and it remains in play. The ball is only out of play when the ball itself goes out, whether on the ground or in the air. This applies in the same way to each of the half, the 23 metre area, and the circle as well.
When the ball is pushed, its location at every instant is relevant. A ball that is dragged or pushed in a constant motion from inside the field of play, outside it, and back in again, has gone out of play. Similarly, a ball dragged or pushed from outside the circle to inside it is now inside the circle; a ball dragged or pushed from inside the circle to outside it has left the circle. If it continued to be dragged or pushed back inside (or outside), it is now inside (or outside) the circle.
This is most often relevant for penalty corners. A dragflick will typically start with a stationary ball outside the circle and finish with a moving ball inside the circle, before it is released. If the ball then enters the goal, subject to usual caveats, a goal is scored.