We don't get Snappers down here & if they do come in they would be rebadged .
\The solenoid will have either 4 or 5 wires on it.
Two heavy ones, from battery & to Starter
One powering the mower attached to the hot terminal on the solenoid and usually fused.
Then either 1 or 2 at the base.
These are your control wires.
In the start position 1 must be +12V and the other ground.
If there is only 1 small wire at the base , it must be + 12V to start and the base itself must be ground.
I don't have a schematic diagram for your mower so any more will just be confusing.