I seem to have converged on a definition of game state, so, for clarity, I'll record it here. This might or might not be the same concept that others have in mind.
Game state is the current state of the game.
It is not really even jargon.
This is the same as the state of a system more generally.
I'll start with a simple example from physics. Consider a single planet or particle (or other point mass) in an otherwise empty universe. The state of the system is the location and velocity (momentum, speed in common parlance) of the particle - these describe the system completely and make it possible to predict its future course.
A more complicated similar system could have several particles, which would have mass and electric charge, as well as locations and velocities. This information describes the system fully and can be used to predict its future behaviour.
(Physicists actually use the state space, which is a space consisting of all the possible states of a given system. https://en.wikipedia.org/wiki/State_space_
A non-physical example is Conway's game of life: you have a square grid with some squares black, others white. At the next step the colour of each square is determined via simple rules by the colours of its neighbours. https://en.wikipedia.org/wiki/Conway's_Game_of_Life
The state of the system in Life is simply the colour of each square.
The physical example had a continuous time, the abstract example discrete time. I will be sloppy and refer to the next game state even with continuous time. The alternatives are more cumbersome and do not add clarity.
The examples above were deterministic. We can add in random factors. The random factor is in the transition from one state to the next.
An example of a discrete time random (or stochastic) system is the "game" snakes and ladders https://en.wikipedia.org/wiki/Snakes_and_Ladders
, a zero player game. The state of the game is the position of the player tokens and information about turn order, and the transition into the next state is rolling a die, moving a token, and going up or down by ladders or snakes, if one happens to a square with one.
A famous continuous time example is Brownian motion: https://en.wikipedia.org/wiki/Brownian_motion
The next state of a random system is no longer defined by its previous state, but the previous state and the random factor do determine the next state.
To get a game, we need to add in players. A game with one player is called a control problem. A game with several is a proper game. Players add their input, or make their moves, as the game transitions from state to state. Games can be random or deterministic. Being "deterministic", in this usage, allows for player input, but there is no random input. (Of course, the players could choose to make their moves randomly. This is fine. But there is no obligatory random input.)
A deterministic control problem is parking a car. The state of the system consists of the position and orientation of the car and its physical properties, as well as the physical properties of the environment. Maybe we can assume there is enough fuel, maybe the amount of fuel is also part of the state. The state of the system and the performance of the driver determine how the system develops.
A player building a GURPS character can also be seen as a control problem. The state is the points that have been spent and what has been bought with them, as well as the total number of available points. The choices of the player, when combined with the state, determine what kind of character gets built.
The state of chess includes the position of all pieces, information about castling and repeated moves, and whose turn it is next. This, together with a player decision, is enough to determine the next game state.
The state of football, a sport where people manipulate a ball mostly by using their feet as the name implies, includes the physical state of the playing field, as well as information about game time, goals scored, fouls and substitutions. (And possibly some other details that I am forgetting now.) When combined with player performance, this is enough to determine how the game continues.
In Burning wheel, the game state includes the parts of the game world that have been determined thus far, the stats of all entities that have been determined thus far, who is the game master, who are the player and which has which character, what has been determined of the game world thus far, is someone in the middle of an ongoing extended conflict mechanism, are there linked rolls whose effect on the following rolls have not been determined yet, and any other resolution whose effect has not yet entered play. And probably other stuff. This, when combined with player performance (including choices) and random input mandated by the rules system, determines how the game progresses.
What about human judgement? Roleplaying games, much like football, have human judgment. This is fine. We can either take the human judgment as part of the rules system, or we can consider anyone making judgments as an additional player. This would mean that the referees in football would count as players. This is a question of modelling and should be done with a purpose in mind - for some purposes we might want to accept the fuzziness of human judgement as fuzziness of the rules system and not worry about it too much, while in other cases we should consider the human fuzziness as essential and important.
What about errors and mistakes? Either they cause the game to stop, they are not critical, or they are fixed in some way to make continuing the game possible. The game has to have a game state in order for it to continue. Sometimes a game state with contradictions or omissions is not critical (humans are flexible in this way) and game can continue regardless, while some of the time the state needs to be seriously discussed.
It might be that the location of a pawn in a game is uncertain (due to the cat), but the game is check-mate regardless, so we can simply ignore the pawn. Or it might be that in one location it is check-mate while in another the game continues, and in this case we'll have to deal with the error in some way. Or maybe it is late and we decide to play another match next day, saying the cat won this one.
What about consent, credibility, and so on? In game studies a game is usually defined as voluntary activity with arbitrary goals and restrictions that the players choose to engage in. (This is different from the definition as a particular type of system, above, but there is lots of overlap.)
This means that any game, in the game studies sense, relies on players accepting the game state, rules, and objectives of play. Playing the game, as an activity, requires this. This is also true of playing in the sense of leikki/lek, unstructured play typical of children, for which English does not really have a distinct word, as well as peli/spel, the structured play which typically has rules and objectives. Roleplaying lives somewhere on that continuum.
In particular, shared imagined space is a subset of game state, since it is used as a factor when determining how play continues. The transition in time from one game state to the next often relies on the system in the Forge sense. Often the credibility has been given ahead of time for some participant to do particular things, possibly privately and without informing anyone else. However, as per the previous point, as a type of play, roleplaying always requires at least passive consent of everyone with respect to everything going on.
Game state: The time-dependent state of a game, which, when combined with player performance and random input, determines how the play continues.