Droping to 10V at start up is border line for dead battery.
As for the rest I need a serial number & model number at least to get a wiring diagram
The only eXmark one I have is for the Laser Z XP which have 3 cylinder petrol or diesel engines
These all use a starting relay and a logic processor which switches the engine on or off .
The FX 730 V each coil will have a single thin wire leading to it
This wire goes to ground to stop the engine
If the key switch has a ground terminal on it then the kill is done at the switch
However eXmark like to use an ignition control relay which is normally closed grounding the magnetos
When energised via the key switch it goes open circuit