I suspect that you have done something wrong.
The starter solenoid is just an electrically operated switch.
A big fat battery positive cable will attach to one big terminal on the solenoid and when the relay (solenoid switch) is operated current is allowed to flow to the starter to turn the engine.
A much smaller wire will carry the control signal to operate the starter solenoid (relay switch) - it will connect to either a separate start button or the start terminal on the ignition switch.
The negative return path back to the battery will very probably be through the chassis of the machine - like it is on most motor vehicles to save the cost of copper wire.
If I read your description of what happened correctly it seems as if you used the starter solenoid as a switch to make a direct connection (a short circuit) between the + and - terminals of the battery - excessive current will have flown for a very brief period.
There will have been high current and heat generated that might - for instance - blow fuses - weld contacts together - melt wires.
It might be wise to have a specialist look at the machine - but you may well have got away with it.
If you carry on yourself - sketch out on the back of an envelope how the wires should go - small control wire to starter solenoid - big power wire from + battery to the solenoid and from there to the starter itself.
The battery -ve terminal usually connects to the frame of the machine.
I know nothing about mowers or engines - no expertise.
Take a second opinion...
April 2015