Yep, I can (could) use a screw driver on the original starter, touching the terminals, to bypass the solenoid and actually start/drive the car. This is what led me to think it was a bad solenoid on the original starter.
Now that I've changed the starter and have the same problem, I suspect a problem with the power delivery to the solenoid. Buying an Ohm-meter today (I know, should have done this already) to check the voltage at the starter.
Synopsis of actions taken to date on this problem:
new battery
new negative battery cable
cleaned terminals (multiple times)
new M3 starter
Any help/suggestions are greatly appreciated, especially if you think I'm going in the wrong direction!!