82 240D ignition engages sporadically

My 82 240D has been a fantastic experience. Bought 18 months ago, drive daily, now has 360,000 miles--love it to pieces.
A few days ago when I tried to start it, I got no response from the
ignition at all, though it had started fine continuously prior to this. I had noticed recently that my automatic transmission had developed a slight hum/buzz in the gearshift console--nothing seemingly problematic, but the gear-shift has always moved very easily between P-R-D-3-2 and, in the past couple of months, it had gradually been having a harder time finding Reverse.
Because of this change in behavior from the gear-shift and the possibly associated hum/buzz, I suspected that the ignition might not be recognizing that the car was in Park and therefore wouldn't fire when the key was turned. On three different occasions now when the car wouldn't start, I have put it in Neutral, rolled it a few feet, then put it back in Park, and the car has started. But this method has failed a few times as well, and is obviously not ideal.
Since then, I have checked connections between battery and ground, battery and starter. I have removed battery connections and cleaned them. I don't have a repair manual, my mechanical abilities could be described as "fairly low but eager to learn." I do have access to a friend's full schematics of the car.
My question is, where is the switch/mechanism that signals to the ignition that the car is in Park? Would such a switch reside in the transmission or the starter or some point in between? (Would it simply be the starter solenoid?) And is my assumption about the ignition not 'recognizing' that the car is in Park at all reasonable? Is there any way to tell whether the car "recognizes" that it's in Park for the purpose of ignition?
After that, I'd guess I'd move on to evaluate the starter and the starter solenoid, though I'm probably in over my head with those, too. I have no idea how old the starter is--it is a Bosch made in Germany.
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There is no ignition system on diesel engines. So does the starter turn the engine every time and does the engine "fire up" only sometimes ? The gear-shift-stick has to be in Park to engage the starter.

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The neutral safety switch is on the left (driver) side of the transmission, about midway between bellhousing and output shaft, just above the pan. However, I suspect it is the shift linkage that goes between the gear shift and the lever that actually tells the transmission what gear to use. The lever is also around the safety switch. The usual culprit is the bushing directly under the gear shift, especially since you hear the humming sound. It is white plastic, about quarter size but much thicker. If the whole linkage is snug, no loose feeling, then the next target is the safety switch.
Take a flashlight and go under the transmission tunnel. It shouldn't be difficult to find the bushing and figure out how to replace it. The problem might be there is not much clearance and you may have to "feel" (instead of see) what you are doing.
Oh, when you go under the car, always use jack stand and block the wheels. Drive-on ramp is OK. Never rely on jack to support the car. Do not trust the parking brake or park in transmission either. The wheels in park may not be touching the ground depending what you are doing. Any mistake is life and death difference.
Jason Weaver wrote:

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One other diagnostic you could try to see if it is a bad switch or loose linkage is to hold the key in the start position and move the gear select back an forth to see if the starter engages.
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Many thanks to you all.
I think the problem may both the missing bushing and the neutral safety switch. I removed the safety switch's harness and connected a wire between its two park/neutral ports (the other two are reverse light ports, I guess) to bypass the switch, and the car started fine, 10x or more. I did notice that there was a lot of play in the shift linkage as Wan-ning Tan described, and it looked to me like there may have been a bushing missing where the two lever parts connect.
I did take the switch off and looked inside to see if there was any grit, etc., but it looked pretty clean. I guess the switch might still be good and just not activating properly due to the missing bushing. Still, I think the switches are not too expensive, and I might just replace them both.
Thanks for the safety tips, too, everybody. Much appreciated.
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Um, if you can figure out to this level, your mechanical abilities is much higher than "fairly low" . Don't underestimate yourself.
The good advantage of old Mercedes is the simplicity. Everything is simple mechanical. Once you know what the wires are, there is no need for fancy tool (not even a code reader).
Welcome to the "club" :-)
Jason Weaver wrote:

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