Any chance you have three examples of your three examples?
Sure.
1) The DM decides that there are 12 goblins in the next room even though he previously wrote down that there are 4.
2) The DM decides that the goblin chief passed his saving throw, even though by the roll he just made, he would not have.
3) The DM decides that a favored NPC is able to take 4 actions on the same turn or is able to do some normally lengthy process as a free action or that a favored NPC is automatically able to move a heavy barrier, even though he would adjudicate a PC attempting to take the same actions differently.
What is the act of a GM Influence that does not have the definition of "Something a player does not like".
Limiting player agency, whether or not the player likes the outcome. I only tend to cite "things a player might not like" because it clarifies why this could be a problem. But the reverse also happens.
1) The DM decides that there are 0 goblins in the next room even though he previously wrote down that there would be 4.
2) The DM decides that the goblin chief failed his saving throw, even though by the roll he just made, he would not passed.
3) The DM decides that a NPC deliberately does stupid actions or attempts things which he is likely to fail at or doesn't exercise powers and capabilities that the NPC has or otherwise is played suboptimal manner in order to ensure the NPC fails. Or the DM decides than a NPC with 40 hit points remaining dies to a blow that did 10, because he just needs the NPC to go down.
This is the same thing but with the GM deciding that he doesn't want to risk the NPC's "winning" at this moment. Honestly though, this line of questioning is to me a red herring, because while there are trends and likelihoods, there isn't something all players agree is the most fun especially when it comes to details like this. There are only general standards that the table expects to adhere to. The situation might be similar to when you are solving an escape room where some people in the group want clues and some don't, and you may even have people agreeing to get a clue that don't prefer it merely to not be perceived as being difficult or anti-social. So focusing on what a hypothetical player in a group may or may not "want" especially in a discussion of Illusionism, which generally involves events that players aren't aware of, is not helpful.
The important thing to focus on IMO is that regardless, the GM's choices to ignore myth, fortune, or rules constitute fiat management of the game, which means that whatever is happening the GM is getting what they "want".
So, here, are you saying each GM must make up every detail of an encounter...and whole setting and make some "offical notes"?
What I'm saying is a good bit more nuanced than that, and anyone who has actually read my discussions of the topic would understand that. What I am saying is that quality preparation (preparing the right things in the right way) is generally preferable to no preparation or bad preparation.
Then the GM must follow the notes they made?
And again, what I'm saying a lot more nuanced than that. What I'm trying to explain is what happens when you don't. I have outlined that GMs are all powerful and as such it's not obvious how players get any agency. Three important ways GMs allow players to have agency is by limiting themselves through rules, by limiting themselves through myth, and by limiting themselves through submitting to the "dice" (or whatever mechanic you use to randomize). GMs can also extend to players direct agency by asking them what happens. Each of those ways is important and has different utility and provides for different aesthetics of play. A GM doesn't have to follow the notes that they made, but if they don't then the game suffers certain consequences and it may be much more artful to follow your notes. Further, I've argued that the more artful and skilled you become, the less temptation you are going to feel to throw out your notes because you've done a better job preparing and you understand better the costs of changing myth mid-game.
So it's fine for a GM to make up anything and make a note last Sunday, but if they do it today it's Always Wrong?
And again, I never said that. What I said that what you made up "last Sunday" is far more likely to be fair and unbiased and not subject to metagaming against the players to get your way, than what you make up in the middle of a session. If you are waiting till the player's proposition to make things up, you never can be sure whether you are being fair because now that you know what the players are actually going to do, you can't be sure if that knowledge is influencing your improvisation of what is there. For example, you might have never intended the mooks to be wearing sentient magical cloaks that attack the players, but after seeing how the mooks went down "too quickly" you decide that the magical cloaks now animate and attack as a complication.
And what about pure Improv? What if the GM has no detailed notes?
Then a lot of different problems crop up that a novice GM will probably be unprepared to handle and may sorts of aesthetics of play won't be fully supported at that time. You'll be winging it, and that will impact how the game plays out. There is a reason most narratives aren't made without some sort of preparation. And GMs need to understand that, and skilled GMs do understand that. Going back to my article on "Techniques for Railroading", if you prepare a detailed plot and timeline of events you may be a railroader. But if you improv everything, you are a railroader. The players might not know. They might not care. But improv heavily overlaps some of the more important techniques for railroading players, and if you are doing pure improv you are almost certainly just laying tracks down ahead of the players. Again, that's fine and you can run a fun game that way, but it's incompatible with certain claims about the game that is being ran.
The "mystery box" does not really work as movies/TV shows are passive and games are active. They don't really fit together.
Of course they do. The "mystery box" technique doesn't depend on whether the narrative is passive or active. Indeed, one aspect of the "mystery box technique", gathering fan theories up and deciding which one is the most interesting and making that one true, benefits from having an active audience. You can of course do it with a TV series because of social media, but its easier to apply in a tabletop RPG which doesn't have sunk production costs and production schedules.
