According to Wikipedia, a magneto starter, also called the ignition magneto, provides current or pulses for the ignition system of a spark-ignition engine, like fuel engines.

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In a nonelectronic small engine ignition system, which of the following ignition components is energized by the power source first?

On things like cars you usually have a belt driven alternator to provide electricity to the ignition and all electric functions. But if you have something simpler, like a chain saw or a small outboard engine, then you can have magnets in/on the flywheel that pass by coils to generate electricity for the spark plugs. And that's a magneto ignition system.

An ignition coil requires external electrical power. A magneto requires only mechanical power to create an ignition spark. If an engine HAS no electrical power (lawnmower, tiller, chain saw) then a coil would not function.