If it was an engine warning there would be a code displayed on the dash along with it usually.
If you leave the seat without setting the parking brake there is an alarm for that.
And on some models in the past if you leave the truck with key on and not running without parking brake set the horn would sound.
What is the full model and serial?
About the 3 wires....
one is power, one is signal for ecm and one is neg

I'm not familiar with that forklift, but......if the truck has been equipped with an aftermarket "loss of engine oil pressure" shutdown system.......that system might be why the horn blows after the ignition switch has been turned on for a number of seconds with the engine not running (or running but sensing a loss of oil pressure). One aftermarket shutdown system I am familiar with is configured to sound the OEM horn a couple of seconds before the shutdown is activated to kill the engine.
Other than that idea, I have nothing else to offer.