2 and 3 is the same for Intimidation (especially if you happen to have the same modifier and DCs).
And what reasons in the fiction are there for there to persuasion to fail but intimidation to work? How do you communicate this in game so that the players understand this and can take this into account?
2 and 3 are different skills, though. If they fail, intimidation can still work.
(In the local lord's manor with the Tarrasque coming)
PC: (knowing the Tarrasque is mythical to the point that almost all believe it not real) "My lord. An army of demons similar to the ones we defeated in Garoka is coming this way. You must evacuate the entire town, because we cannot face them before morning."
DM: "Roll deception."
Player: "2 on the die for a total of 10."
Lord Skeptic: "How dare you lie to me! My steward is practiced in the arts and he would know of such an invasion. And I can sense the falsehood on your lips."
PC: "You are right. I lied because I feared the truth would be even less believable..
Lord Skeptic: (interrupting) "And what is this truth?"
PC: "...the Tarrasque is real and it is coming this way. Nothing can stand in its path. My companions and I might be able to stop it, but as I said, not before morning. We have to have rest or we will surely die in the attempt. Please! Please evacuate before it's too late."
DM: "Give me a persuasion check at disadvantage because of the lie you were caught in."
Player: "Dang it all! Natural 1 for a total of 9."
Lord Skeptic: "You expect me to believe in fairy tales now!? Enough of your lies. Get out of my sight before I have you arrested."
PC: (growing angry): "You were right to name us as the ones who freed the country of Garoka from the hordes invading from the Abyss. We are also the ones who defeated the great dragon of the north, leaving its carcass to rot in an empty field. And we stopped the god Snee Kee from turning the city of Suhkurs into a bastion for thieves and assassins. If you don't evacuate this city now, by the gods we will track down you and your family, executing one for every one of your people who dies here. Evacuate, now!"
DM: "Roll intimidation."
Player: "Finally! 27!"
DM: "Lord Skeptic turns sheet white at your threat. He knows you are capable of making good on it. After a few moments to process he responds to you."
Lord Skeptic: "F-f-fine! But know that I will never forget this. You may be powerful, but I am still lord of this place and your better."
(sometime later with half the town destroyed and the carcass of the Tarrasque at the feet of the PCs)
DM: "You see Lord Skeptic walking towards you nervously."
Lord Skeptic: "I... (he glances towards the Tarrasque) I never imagined that it could be real. You saved countless lives here. (Lord Skeptic fidgets a bit) I will not.....................seek any retribution for your threats. You had no other choice. I can see that now. In fact...(Lord Skeptic turns to his scribe)... Chibbers, take note that these heroes are welcome in my town free of any duty. I will pay the inns and stores for what they wish to buy, within reason."
As you can see, the three skills are very different in what they are trying to accomplish and when you might want to use them. Failing at one might impact another, but wouldn't preclude another from working. They aren't the same. Nor will intimidation always result in bad stuff happening to the group, even if the person being intimidated is rightly angry and/or afraid of the PC(s).