You don’t see people wearing spurs much any more, which may explain why some are vague about the significance of metaphorical spurs. Anything that prompts you to do something can be a spur to action. We say of people that are prompted in this way that they are “spurred on” by fear, ambition, greed, or some other cause.
So a momentary impulse which causes you to act without advance planning can result in a decision made “on the spur of the moment.”
Then there is the expression “spare moment”: “When I was getting the kids ready for school I couldn’t spare a moment to clean up the mess the dog made in the kitchen; would you please do it?”
This latter pattern seems to lead some people to mistakenly imagine that the expression is “on the spare of the moment.”