The blanket claim that cheating disables achievements is close enough to true to be repeated constantly, and wrong often enough to be worth checking per game.
Blocked for the session
Grand Theft Auto V takes this approach. Using a cheat disables achievements while it is active, and the block lifts when you reload without cheats. A single cheat does not permanently damage a save.
Blocked permanently for that world
Minecraft flags the world itself. Enable cheats and achievements are off for that world forever — turning them back off does not restore them. The workaround is a separate world, not a separate setting.
Blocked, but the community fixed it
Skyrim disables Steam achievements once the console is used. Mods exist that restore them, and they are widely used.
Not blocked at all
The Sims 4 has no achievement penalty for cheating. Nothing is locked and no progress is invalidated. This is unusual and worth knowing if you assumed otherwise and have been avoiding cheats for that reason.
Why there is no single rule
Achievement systems are implemented per game by the developer. There is no platform-level standard requiring any particular behaviour, so each studio decides — and older games often decided before achievement culture existed at all.
How to check safely
Make a manual save first, use one cheat, and see whether the achievement indicator still appears. That takes two minutes and beats trusting any list, including this one.