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Guest 6801328
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I think there is a misunderstanding here. What @Bawylie is saying is that no matter how it is presented by the players, as long as the goal and approach is the same, the DC is the same. So let's say you have the Cha-10 character's player give a first-person in-depth reasoning for why the king would benefit from lending the vanguard to the dance competition. Tears are in the eyes of everyone else at the table after this thespian finishes his or her speech. In an alternate reality where the same situation is playing out at the table, a Cha-20 character's player just says "Hex Arcana tries to convince the king to lend the vanguard to the dance competition because it will show a friendlier side of the guard to the people and improve their reputation."
The resulting uncertainty is the same as is the difficulty since the Cha-10 character's player has said the same thing with more (and perhaps more stirring) words. The DM is judging the goal and approach, not the word count or acting ability of the player. So it does matter what you say. It does not matter how you say it, provided you have at least said what you want to do and how you want to do it. (Although it might matter in another way, such as if you might earn Inspiration by using flowery speech or perhaps by being blunt and to the point.)
"I persuade the king, Persuasion X" is insufficient here because the DM cannot judge the goal and approach and decide if there is uncertainty, a meaningful consequence of failure, or set a DC. Further, the "lead up" would never be contradicted by the roll in the method described above since the roll determines the outcome - the king does or does not approve the request (or approves it at a cost or with a setback for the PCs).
If you don't mind me adding to this excellent example...
I hope, but don't require, that in both cases the player brings in their character's personality, whether that means using the formal bond/trait/flaw/ideal or just the persona and quirks that they've developed for this character. And that could be done in "acted" 1st person, un-acted 1st person, or 3rd person.
But that's just for the enjoyment of everybody at the table. It's not going to effect the mechanical difficulty of the task they are undertaking.