Wow, this is a way out in left field guess, and it's probably not what is going on with that vehicle. That said, sometimes the designers of vehicles have built into the system a way to make the spark plugs get much more power while starting and maybe for a little bit after that. So, I'm thinking that it might be drawing a lot more amperage for that brief amount of time. However, the fact that you're getting some kind of a buzzing noise might be that you also have poor contacts and they can't handle the extra current for that brief amount of time.
If you know anybody with a clamp on amp meter and can isolate the wires that are being energized from that switch it might be worth your while to take some readings and see if indeed there is at least one wire that has excess current going through it for that time that you're getting the buzzing. The excess current might be normal but that would indicate that maybe the contacts of the switch are worn out they can't handle it also because of their buzzing.
Again, this is still a way out in left field guess.
However, years ago, like many years ago I do remember that ther were vehicles that when I got into what was going on with them during start they had More Voltage going to the ignition coil and after start they ran the ignition coil at less voltage going into it. They actually had dropping resistors and during start the dropping resistor that was between the Plus of the battery and the power for the ignition coil was a much lower value allowing higher voltage and more current to be applied to the ignition coil power input circuit.
One of the reasons any engine manufacturer would want to do a hotter spark during start and for a brief time after is that during start you're running the engine really rich and you combine that with cold weather (cold air being drawn in and a cold engine) and you have a great recipe for getting the spark plugs so wet that they actually won't Spark because the fuel that has them so wet can actually short them out. So the recipe for preventing that from ever happening is to have the spark plugs running extremely hot for a very brief amount of time during starting, and maybe for a little bit more time until the fuel system has stabilized to a rich mixture that is not rich enough to wet a spark plug. Also, the temperature of the spark plug comes up pretty quick and after a half a minute or so it might be that the problem of having it get too wet becomes a lot less likely.