The TVTropes Trope Finder is where you can come to ask questions like "Do we have this one?" and "What's the trope about...?" Trying to rediscover a long lost show or other medium but need a little help? Head to Media Finder and try your luck there. Want to propose a new trope? You should be over at You Know, That Thing Where.
Well, there's Friendly Zombie.
Currently Working On: Incorruptible Pure Pureness
^ overlaps with Revenant Zombie when they also retain their minds
So do I make a YKTTW as a related trope to Friendly Zombie and Revenant Zombie?
These two tropes thoroughly cover the first form you mention, and I had forgotten Undead Laborers for part of the second.
That said, if you're looking for a project I think there's room for a general trope about taking a dangerous monster and rendering it (at least mostly) harmless for fun or profit, ala Pet Monstrosity, Undead Laborers, or Mainlining the Monster.
The Zombie Apocalypse came and went... and it turned out pretty well. Apparently zombies just weren't that dangerous to begin with, or alternatively, someone figured out a way to neutralize the threat from a zombie, or maybe from ALL zombies.
This trope has two variants.
-The zombies in the setting just aren't really dangerous to begin with. They're not aggressive, they don't bite, they don't infect people.
—Example from Roadside Picnic: the main character's father, who died before the story started and was buried in what would become the Zone, eventually comes back to life thanks to the Zone's weird powers, and he goes back home and just kinds of wanders around like a senile grandpa, he occasionally talks and he eats at their table and doesn't pose any kind of threat. Instead of being a decrepit zombie, he apparently radiates good health
—Example from The Sims 2: Zombies can be brought to life (either by trying to pay the Grim Reaper for a resurrection but being too cheap, a Genie wish backfiring or an evil spell). The zombies don't bite Sims and spread zombiness. They just groan and shamble around, lose their personality, and inspire disgust, but they can learn skills, take jobs, have relationships, and just keep on (un)living. And they're immortal.
-Zombies are dangerous, but someone figured how to make one harmless and even useful. This can be on a small scale (a survivor removes the jaw and arms of a zombie so it can't grab or bite) or a large one (all zombies have their jaws removed and are used as menial labor).
—Example from Shaun of the Dead: after the zombie apocalypse is dealt with quickly and efficiently, the zombies are given muzzles and used for things like sweeping the floor, operating checkout lines, or in game shows for entertainment.
—Example from Fido: There was a zombie apocalypse thanks to magical space radiation, but things stayed under control and zombies are outfitted with collars that make them docile and people keep them in their houses for, once again, menial labor.
Edited by Commander_Ysenir