Mine doesn't have a reset button that you push. It has a knob that you rotate which 'collects' all the trip numbers and aligns them. You stop rotating when everything reaches 0.
Although I've taken mine apart and inspected it, you bring up an interesting point about resetting while the car is in motion. I know the problem with the drive wheel slipping can not be fixed by a simple press of the reset button, but perhaps we could have prevented loosening the wheel by _not_ resetting the car while in motion. I'll have to look at that next time.
Anyway, congratulations on your fix. It's always good to hear DIY'ers avoiding dealers trying to rip you off.