If you don't press the horn button and instead press the starter button does it crank and start without the red light being on?
I suspect the horn, assuming it is actually making a noise, produces vibrations causing an intermittent open to close.
Could be a loose Ign bulb connection.
Could be something more complex.
Could be the Ign switch itself too.
The switches on these, including the hand controls are not weather friendly and can stand periodic cleaning.
An el cheapo VOM, a decent set of leads and an afternoon can identify whatever it is.
All the best.