Haha, well hell... who knows what the "correct" was is, or if there is a correct way. It would have been years ago, but I remember reading something about the supposed best way to remove the stem. What I remember reading was that its supposedly having the stem pulled to time set moved the keyless to where it was more difficult to dislodge.The funny thing is, I ALWAYS remove the stem in the winding position and I had never a problem with the keyless.
Most important thing is using a 1.00 screwdriver to push the release button. You can't push the button to deep with a 1.00 screw driver.
Turn the stem gently when you put it back in the movement.
And @Thrasher, my comment wasnt directed at you, but more at the idea that cheap ETA parts are the issue.