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Not sure what two wires you are referring to. Is it the large duplex battery cable? There shouldn't be any buzzing from the engine. When you refer to turning the key to ON do you mean just ON or START?
Always use a new battery.
Check the fuse at the engine, check on the start solenoid where the battery cable attaches for 12 volts. Use a 12 volt test light or volt meter. If you have voltage there check the small black wire to the start solenoid goes to ground. Check the brown or yellow/green small wire on the solenoid it has power when the key is turned to start. If no power at the solenoid check the neutral start switch in the control.
The picture you attached, where on the boat is this?