It’s a tough question to answer. Both yes and no.
It really comes down to the narrative “weight” placed by the story / the players on the scene. The less important the scene, the more likely its to be resolved with a more abstracted check, while the more important the scene the more detail goes into our descriptions.
So, yes, the negotiation and treating trollblight would be handled the same. I do ask for players engaged in an “important” negotiation scene to — even if not acting in character — to describe their overall intent and approach. What do you want from the NPC? What are you offering to the NPC? And do you have a particular strategy (e.g. drawing on a shared background with NPC, using something the NPC just said to trap them into agreeing, playing a tune to pull on their heartstrings, intimidating them with threat of an approaching monster, etc). In the example I cited, the player said I “Help” without further explanation so I ruled that he could not.
With the treating trollblight...assuming it carries heavy narrative weight...maybe there’s steps involved — identifying that its trollblight, finding patient zero / infected troll, acquiring sample, researching an antidote, brewing antidote, and applying it to the victim. So I’d possibly — depending on specifics, this is all theoretical — expect a PC wanting to Help another with treating trollblight to declare where in that process they are helping, and generally how they’re assisting (soothing patients while samples are collected? managing the distillation & titration process? wrestling an infected troll?).
And no, they’re a bit different by necessity because D&D involves actual dialogue as one of the ways players have fun. Interaction, along with combat & exploration, are pillars of the game. Always have been. No special knowledge is required to do a bit of roleplaying or state your character’s intent & approach. Very possibly I wouldn’t require a roll at all if the player’s position in negotiation was strong enough. In that case there’s no a Persuasion skill, it’s just roleplaying. DM decides whether check is necessary, as per 5e rules. Whereas getting into descriptions of specific skill uses — horseback riding, or wilderness survival, or medieval / Renaissance-era medicine — is not at the heart of the game, it’s not one of the 3 pillars. Certainly, it can add a lot of flavor, and a savvy player might coax a DM into granting advantage for a vivid description, but there’s no expectation built into the game that players are going to, for example, know how to treat a wound, build a shelter, or take a wild horse.