It might well be worth looking at the system from Exalted 3rd Edition for inspiration actually, it does a lot of things right.
Characters have 'Intimacies', these can be 'Ties' or 'Principles'.
A tie might be 'My Family (Love)', 'Evil Prince Bob (Hatred)', 'Duke Jim (Wary Respect)', etc.
A principle could be 'Honesty above all', 'The Faith of the Holy Juggling Balls is the true Path', 'Everyone is only out for themselves', etc.
Each can be Minor, Major or Defining. Minor are still significant parts of a character's worldview but tend to only come up if directly relevant, Major intimacies come into play even if circumstances are tangential, Defining intimacies are the foundations of your worldview and the kind of things your character would lay down their life to protect or further.
How does this work mechanically? Well they are tied into willpower recovery as per White Wolf/Onyx Path standard but they are also what social influence is driven by. If you try to persuade somebody of something then you get (significant) bonuses or penalties depending upon how it touches upon intimacies. Trying to persuade somebody to go against a Major, let alone Defining, intimacy is almost impossible, with huge bonuses to their resistance rolls, persuading somebody of something that aligns their one of their intimacies is somewhat easier however (though the modifier this way is smaller).
That said, social influence is rarely about 'making' people do things, it is about discovering then trying to influence their intimacies. So you do not just walk up to Jill, who legitimately loves Bob (Major intimacy), then try to get them to sleep with you. You cannot even erode Jill's intimacy from a blank slate start, you first need to build at least a minor intimacy that would support such an erosion (Or a Major intimacy to try to weaken a Defining one). Strengthening intimacies beyond Minor then has a 'burden of proof', (or at least convincing lies). So you might persuade somebody that the Duke cannot be trusted as a Minor intimacy, after telling them of how the Duke abused hospitality when visiting your castle. To raise this to a Major intimacy you would need something more serious/convincing, perhaps introduce this person to a trustworthy seeming Baron the Duke once betrayed.
Persuading somebody to do something as opposed to trying to influence their beliefs requires you to successfully play to an existing intimacy (positive or negative) and the degree of what they might be potentially willing to do depends upon the level of said intimacy. A minor intimacy could let you persuade people to do something inconvenient but not life disrupting (They Respect Honesty, persuade them to go to the City Watch and give an accurate report of what they witnessed). A major intimacy can be leveraged into more significant actions (Major Intimacy that the Faith of the Holy Juggling Balls is the true path? Join my mob, we are going to to tear down the false temple of the Church of the Holy Bowling Balls!). A defining intimacy could help persuade somebody to join you on a suicide mission, donate a large portion of their fortune to your cult, etc. (You hate Evil Prince Bob? Here, take this knife and stab him during the ceremony.)
Without intimacies directly supporting a social influence effect you have to bribe/bargain/pay somebody, or threaten them. Even if successful a threaten action only gets somebody to do something for that scene and also gives them an immediate negative intimacy toward you along with weakening positive ones.
You can also try to 'Inspire', which lets you say make a crowd angry with a rousing speech or similar, but the target player specifically decides how they act upon this and does not have to drop everything immediately, just be genuinely resolved to action. If you fill somebody with sorrow, they might go away and cry about their dead wife, or they might donate money to charity, or go on a week long trip to visit their grandchildren to cheer up. It is treated on the same level as a Major intimacy for as long as they act on it.
Keep in mind that intimacies can be significant bonuses, they give benefits to furthering them, etc, so this really comes down to most social influence being about persuading characters to do things they already value.
Added: To use the intimidate example, if somebody has a Major Intimacy about not showing fear then that might well make them really, really hard to make back down through threats of violence even if their social resistance stats are mediocre, but it would probably also make it really easy to goad them into accepting unwise duels or similar shenanigans. Also they would need to actually define that as a key part of their character.
