I have a lawnboy10546 ser 7904307 with electric start, it won't start woth the electric start, but one pull with the cord and it starts right up. the electric start spins the flywheel thw same as pulling the cord. but it will not even attempt to start. does that bosh plug by the starter or that thing that looks like a stater on the flywheel have anything to do with the electric start. its not the battery before you ask. it spins fine.
That stator thing "on the flywheel" is a little alternator to charge up the battery while running. Based on my inspection of the circuits in the lawnboy repair books, the ignition system on 2-cycle, electric start, walk-behind lawnboys is entirely separate from the starter wiring. When you squeeze the bail on the handlebar closed it un-grounds the coil, allowing for spark, and when you pull the rope, or engage the starter it should start equally well. The method of spinning the flywheel (rope or starter) should have nothing to do with the engine ignition. Nor is there any other difference I can tell. All the electricity to provide the spark is made from the ignition coil and magnet imbedded in flywheel.
I believe the relay is to switch the battery over from sending current to the starter, and receiving current from the alternator for charging back up. Again not relevant to the ignition circuit which is totally separate.
Here is a trick I found that works about 75% of the time with electric start push mowers. If you have a primer, prime 4 times. Slowly pull the starter rope all the way out and return. (You are not trying to start the engine, just getting a liitle fuel flowing) Try the electric start. Believe it or not, but manual starting turns the engine over faster than the electric starter, making it easier to start.