I think the excess movement is in your steering system, but with the engine off you are not moving the steering wheel enough to have it show up.
Try lifting the front of your car so the wheels are off the ground and then try it with the engine off. Have someone move the steering wheel while you inspect every link and connection.
Often quick rocking back and forth of the steering wheel will make the steering system report any loose pieces.
Check the idler arm bushings, tie rod ends and also your steering dampener. The dampener can easily be checked by removing one end and pushing and pulling on it.
If you are wondering about your pump, try changing the fluid and also replacing its filter.
Keep us posted,
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'65 911 Coupe (#302580)