1. "So I want to convince him to open the gate."
"What do you say?"
"Uh...Hey sailor, I'm hot, open the gate!"
"Roll charisma."
2. "So I want to convince him to open the gate."
"What do you say?"
"I'll give you 300 gold pieces if you open the gate."
"'I must admit to being intrigued' he says 'half now, if you wouldn't mind...'"
"Nope, only after you open the gate."
"'And why can you not fork half now?'"
"Well if you're lying I...uh...I won't be able to eat if I can't get past the gate and don't have any money, whereas either one alone would be sufficient to make sure I survive the night."
"Roll charisma."
3. "So I want to convince him to open the gate."
"What do you say?"
"'Open the gate now or when I do get through we'll kill you and your whole family and mom'"
"'My mother is in my family but the over-arching point is it doesn't seem like you're getting past if I don't open the gate'."
"Oh are you sure? Is every checkpoint between here and Dwindowndale as well protected as this one? Are all the guards well rested? Is every sentry as dedicated as you? Is every guardhouse built as solidly?"
"Roll charisma."
4. "So I want to convince him to open the gate."
"What do you say?"
"Open or we kill you now."
"The guard looks through the slit at you and sizes you up.--Roll charisma."
5. "So I want to befriend the guard in case we have to get back through the gate later."
"Ok what do you say?"
"I say, good man, the weather is surpassingly mild this day is it not?"
"It would appear to be."
"What have you got there?"
"A marmot."
"It is a fascinating beast! Might I examine it?"
"Roll charisma."
____
Ok so these are all ways I'd use Charisma in D&D. If I was using Persuade in Call of Cthulhu I'd use it the same way. There's a specific method here, but it's discretionary:
Players can say what they want and make whatever arguments they like and may threaten or cajole, but basically the NPC always has the same choice:
-decide whether they trust what PC is asserting or implying (and then take action based on that decision), or
-decide to keep talking.
If they decide to keep talking, they keep talking, if they decide to make a decision, that's when the dice take over.
And which one they decide to do at any moment is totally based on how the GM is playing the NPC. Basically, if you keep talking, you delay the roll and give the player an opportunity to modify it, for good or ill, based on what exactly the player says.
Unless an actual action happens (throwing the bribe to the NPC, attacking, giving up, deciding s/he doesn't want the thing any more) the NPC will not do anything until there's eventually a Charisma roll. The conversation is (underneath) about deciding how to modify that roll. Just making the NPC laugh could modify the roll. Offending the NPC could modify the roll. Anything that happens before the dice hit the table might modify that roll.
Once the roll happens, it means the NPC has decided some fact ("these people are a threat""this girl is hot") is true or not true and new information will have to be put into play before that can change.
In example 1, exactly what the PC wants the NPC to believe is a little nebulous but it's basically something like "1. I'm attractive 2. You letting the gate down will allow access to attractive me on some level". If the PC were accompanied by an obvious partner (riding on the same horse as someone of the same social class who had their hands around his/her waist) then either the guard would respond and point that out or I'd just make the roll straight with a penalty depending totally arbitrarily on what I though the guard was like.
In example 2, the question is, at first 'Does the guard believe s/he's actually going to get the money?' Now depending on what guard I'm playing I could've called for a roll right after the third line (that's how a greedy but stupid and quiet one would've done it), but this guard is more canny. I could've called for a roll after the 5th line, but I didn't and kept the conversation going. By the 7th line, the PC has pretty much pushed the conversation to a point where the NPC I'm imagining can't go any further without deciding whether the PC is telling the truth, so the charisma roll comes. But it could've kept going, like: "If you are unfamiliar with these lands, you'll find there are many beasts of the field hidden in the tall grass, ye may not starve as quickly as ye think!"
Example 3 is much the same thing, though depending on factors already established about the setting, there may be modifiers on this roll. Like if the PCs know that some of the checkpoints are in disrepair or that many of the guards are known to be slackers. Point is: the PCs argument has made these factors relevant to the roll.
Example 4 has charisma basically used as an Intimidate roll. Do the PCs look like a credible threat? If the PCs are carrying heads on poles or the wizard has a purple worm for a familiar all these things might modify the roll even if the PCs didn't mention it.
In example 5 it looks like the PCs have pushed the NPC to a decision point: Hand over the marmot or don't. Really the NPC has, of course, three choices: hand over the marmot or keep talking or roll initiative.
A key point here is if the befriending works, the reward is a prejudice in the PCs favor later. In the other conversations, the goal is right there (open the gate), in this one, what specific thing do the PCs want the NPC to believe overall? They want the NPC to believe "There is something to be gained by staying on this marmot-fondling cleric's good side". If the conversation is successful, any other rolls involving that guard have to take that into account.
Here's an important bit: the persuasion in the game is not based so much on the players' ability to act verbally in a convincing way like an actor when playing the PC, but to (quickly) make arguments and suggest implications that would convince the person in the fiction. The charisma stat covers actual undefinable charm. Much as the player states the arrow's target and time of release but the dice and the PC's dex stat determines whether it's a hit. This uses Charisma the same way D&D and other older game use every other stat: the player decides the tactics to the degree that they can without actually being in the fiction, the PCs stats and dice decide the quality of the execution of those tactics.
(Compare and contrast with the way other systems use "social combat" in much more abstract ways.)
None of this takes into account just what the specific mechanic on the charisma roll is--it can be done a lot of ways: a straight roll a roll vs wisdom, a roll with a specific skill (intimidate, diplomacy etc), it doesn't matter which.