I'm sure BP's description has made things clearer. What I suspect you are not noticing is the forward movement before the backward.
As to the handle, its interaction is very simple: it just pushes down on the knob shown in the photo below (H).
Do not adjust anything till you are sure it's wrong, but take note of the coin detection rod (CD). This is adjustable with the bolt holding it in place and moves forwards and backwards to give just the right clearance to allow the coin to drop into its holding place when inserted. It could be the wrong setting for the CD that is stopping you from seeing the forward movement. When at rest, the rod should be just clear of the coin drop to allow the coin to stop in front of it, for the forward movement will not be very great. Remember, although the arm tries to move forward, it's the lack of forward movement that unlocks the machine, which is what you are seeing when you exaggerate the movement of "E".
Referring to your 3 photos:
In photo 1, the machine is, as you say, locked and won't play. This is because "E" has tried to move forward and can, because there is no coin in the way, so no play.
In photo 2, you have simulated what happens when a coin stops "E" from moving forward and unlocked the machine. All you need to do is adjust the CD so it stops "E" moving forward just enough to unlock the machine. As BP says, it's a small sweet spot to find.