In a magneto, what does the term "coil core" mean?

Answer 1

The heart of that part of the machine.

According to Wikipedia: http://en.wikipedia.org/wiki/Magneto_%28electrical%29
"A magneto provides pulses of electrical power to the spark plugs in some gasoline-powered internal combustion engines where batteries are not available, most commonly those in 2-stroke and 4-stroke engines used in small motorcycles, lawnmowers and chainsaws."

The electrical energy from the generator needs to be increased to make the spark the COIL CORE does that. The coil core acts as a transformer that reduces the amps and increases the voltage to make the spark hotter. Otherwise the engine won't run, the cycle fail and the engine will grind to a halt.

A battery can be used to keep it running, but that will soon drain the battery and kill it along with the engine.