Discourse is persuasion by argument: evidence marshalled, objections anticipated, a conclusion built where the listener can watch it being built. It is the slowest of the persuasion skills and the most durable, because a listener who has been argued into a position generally stays there — and unlike Charm or Guile, it works on people who are actively suspicious of the speaker.
Discourse is what carries a formal case. Arguing a matter of law aloud is a Discourse (Law) test: the law supplies what must be said, and Discourse decides whether saying it works.
Debate. A contest between two speakers is resolved with opposed Discourse Success Value tests, each taking the skill relevant to the subject as a Secondary Modifier — Discourse (Folklore) on a question of history, Discourse (Melee) on a question of arms. Whoever reaches the higher Success Value has made the more compelling case to an impartial and knowledgeable judge, which is a narrower audience than a debater usually gets.