I have both digital and analog servos on many planes.

If I turn on my plane and radio, it centers the analog servos.

Then if I manually push on any surface to change the position of the control surface, thereby moving the analog servo...

IT MOVES RIGHT BACK TO CENTER.
Thats exactly what I was saying... you can pushback an analog servo. You cannot pushback a digital servo without it increasing power to try to hold its position. An analog will give, where a digital of equal torque will not.
Likewise if I hold it at a particular position and move the surface, it goes right back to that position.
Again correct, but a digital will not let you move it,,, it will "push back against you increasingly" to hold its position, an analog will not pushback against you increasingly, it will push a certain amount and thats it, after that it will give in and allow itself to be pushed (by air movement or your finger).

As soon as there is any force moving the control surface away from the commanded servo position, the analog servo starts pushing the surface back to where it was.
Yes, as AFTER the force is removed, but it cannot overcome the force once it starts to push the servo back, to overcome the pushback. You can increase the stick movement and possibly overcome the pushback if your linkage is setup to give you increasing mechanical advantage as throw increases. A digital will increase power and resist being forced out of its position period.



The servo want to go back to a particular position.
No an analog servo does not WANT to do anything, it just recieves power and does whatever it does.. A digital WANTS to satisfy your stick input and hold its position and it will do so by increasing power to hold its position. An analog wont.