The electronic component is probably not a resistor, but a DIODE.
The wire running back to the coil is the power used to excite the alternator at startup. The coil is just a convenient source of switched 12V power in this case.
It needs a diode to prevent backfeeding the coil from the alternator. That's why it won't shut off with the key.
Install the diode with the "arrow" pointing toward the alternator.
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Today's Featured Article - Identifying Tractor Noises - by Curtis Von Fange. Listening To Your Tractor : Part 3 - In this series we are continuing to learn the fine art of listening to our tractor in hopes of keeping it running longer. One particularly important facet is to hear and identify the particular noises that our
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