BAD RONALD
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Could the previous owner have installed a remote starter?
Sounds exactly like a remote starter. Raise the hood and look for a warning decal stating "equipped with remote start, disconnect battery before servicing" or the like.
My '04 E350 is remote start equipped. Bought it used in '10 and drove it for almost four months before I figured that's what the fourth fob button was for. On mine it is blue with a key icon on it. Stepping on the brake pedal without the key in the ignition and turned to the on position will kill the engine as you describe.