A scoring system is a quantitative evaluation, designed to enter a singular verdict into some official record.
Short version: a scoring system is a system designed to produce convergence of judgments. If you agree to the scoring system in the beginning, you’ll agree to the final judgment it produces.
The guiding thought: scoring systems are designed to produce a singular verdict.
Note that you can have a game or a competition without a singular verdict. Skateboarders can compete for the coolest trick with a plurality of senses of cool, and a plurality of judgments.
In the history of skateboarding, it was the professionalization of skateboarding and large prizes that led to the need for singular judgments. To do that, pro skateboarding started to focus on measurable outcomes — height and number of flips — over aesthetic qualities.