If this is the original radio. I believe it has an electrical/electronic latching system that turns it on/off. Since I assume the memory for the preset station is retained It would almost have to be some reaction between the main power feed and the memory retention feed. That is they are interacting somehow because of a flawed ground perhaps.
Anyways the first thing I would try is running a temporary external ground to the radios metal case. In fact I might even run that temporary ground right back to the negative terminal of the batery. For the time it takes it might prove worthwhile.
I was just wondering if it could be done without pulling the radio. Although I cannot believe you are asking on this site if it where that simple.
Hopefully when the cause for one deficiency is found the other will stop as well. Pick your poison. The only other possibility might be the main power wire is going to a different potential for just a milli second as you turn the keyswitch. The potential difference between the always on memory retention feed and the main feed might just trigger that latching system in the radio on. Under normal circumstances there is no difference in potential between the two feeds or too little to matter.
One thing for certain is there are no little elves from the black forest stomping around inside that dash pushing your buttons. Hoping someone else on sight can give better ideals. You have helped so many people over the years I hope they can help.. If there is any difference like an aftermarket radio or whatever from normal it should be mentioned.