There are design choices that are make the player more successful in the game and doing it for they enjoyment.
One of them is bending/manipulating the rules in the interest of victory of the player.
This practice is completely legal and done by your interests in mind. That is not a criticism of the games. But I would like to know when it is happening. (This is not a question of right or wrong. If you want to talk about that please open an other discussion!)

Do you know games like these? What does the actual game do when it cheats for you? Do you know a way/option to turn it off?

Please try to avoid SPOILERS about the games(story or levels or manipulation of player psyche... etc). If you have to mention something about them USE SPOILERS WARNINGS!
Discuss only the mechanics please.
If you have official sources you are welcome to do so.

I give you some (well known) starting examples:

XCOM Franchise:
Mechanics:
A lower hit chance (percentage) is shown instead of the true value when the player makes decision or selects it's target to shot an alien.
Source:
www.gamasutra.com/view/news/266891/Jake_Solomon_explains_the_careful_use_of_randomness_in_XCOM_2.php
Known solutions:
It can be turned off by setting to the highest difficulty level. Also there are mods to solve this.

Hellblade: Senua's Sacrifice:
Mechanics:
<SPOILERS
the infection never lethal, no Permadeath (this is the manipulation of player psyche hence the spoiler warnings)
SPOILERS>
Source:
www.youtube.com/watch?v=xZsJkDZiajU
Known solutions:
It cannot be turned off.

Amnesia: The Dark Descent:
<SPOILERS
the madness never lethal
SPOILERS>
Known solutions:
It cannot be turned off.

(There are games that use the "Coyote Time" when the player is still considered to be on a platform while their enemies aren't. Feel free to name these if you can.)

Thank you

In BioShock, the first shot from an enemy always misses.

Also, a very experienced game designer told me that whenever you see a percent change of anything to succeed, it is usually lower than the real chance, so the player feels better because he "beat the odds". But I see you already mentioned that in your post.