I don't necessarily disagree with any of this, but where you talk about people being bound by the rules and seem to mean what people generally do when they refer to rules -- a set of formal, written strictures -- the "rules" that I aim to adhere to and consistently apply are more abstract -- they're an understanding, shared with the players, as to how the world works. The physics of the fictional world, if you will, and the game rules are secondary to our understanding of the fictional physics.
On this I agree and think we are aligned. There is an order within the game, the rules themselves and the game we have crafted with them, that should be adhered to in order to execute the game consistently in a way that doesn't feel arbitrary. I care for systems with some amount of concreteness --- if the character is attempting X, that's adjudicated this specific way and we abide by the results, the physics --- and extend that logic up a layer --- if this situation is to occur, it's because these previous actions have logical reactions, the understanding layer. I think I would agree that this amount of consistency (both adjudication and "meta-adjudication" let's say) is closely aligned to impartiality.
Where impartiality comes into (for me) it is that my world doesn't treat PCs as inherently special in any way just because they're PCs. (Naturally, if we're playing Godbound, they're special by dint of the fact that they're demi-gods, but in such a case they're not inherently more special than any other demi-god. The same general principle applies.) The outcome of a situation isn't going to change based on my decision about what would makes for a compelling story arc, it will change because the existing logic by which the world operates indicates it's a logical one. A character who is fascinated with clocks isn't more likely to come across clocks than anyone else (instead, I would expect such a character to make more of an effort to find clock than others). I don't adjust a fight to be easier or harder based on my feelings or my player's as to what would be "fair". If my players want something, I expect them to go get it.
To me, that's the crux of the partial/impartial divide -- do the PCs get special treatment or favour? Does the world adjust itself to the needs of the PCs or the story, or are the players/PCs limited to acting through their PCs and getting the outcomes any character could in the same situation?
And here is the delta. I should lay out that I prefer games, or houserules in games that don't have what I'm about to describe, where the players
themselves have a role in adjusting the game that is beyond taking and resolving the actions of their characters. For example, 13th Age icon connections, where the world adjusts to the players' ideas, with only minor adjudication from me, because the player spent a point of story metacurrency, and I go "haha, yes... YES!" as I scratch out some notes in my brain and think about the new truth of the game.
I don't know if I consider that situation as the PCs, specifically, having special treatment or favor, but in essence that's how it plays out --- these events, whatever they are, happen because these specific in-game people
are guided by a different set of physics+understanding that comes from out-of-world.
With or without that metacurrency spend, perhaps where I am is that there is a default logic, but I still find that I prefer to override it. I don't think that can happen if I am impartial --- I would make those story arc moves, I would "somehow" get the game into more clock-having situations, and so on. I think part of our approach delta is that I allow that to be explained retroactively. The icon connection or my whims (influenced by the players or not) may first land in the game, and the game understanding gets retrofit to "justify" it second. That also feels afield of logical arbitration, and when I GM, I am retrofitting things often.
But, that's what I like. I pick these systems and approaches where the players/PCs get to do special things beyond character actions, and I put my thumb on the scale. Your approach is completely valid and it sounds great for you, too. I just can't pretend my style is an impartial one.
