AbdulAlhazred
Legend
Right, so I think we CAN talk about classifications of agency with respect to specific ELEMENTS of a game. That is, the game is a progression of states (@Manbearcat described this in his last post above). There are 'conditions' which must appertain to a transition from one such state to another. Players can have agency over those transitions by virtue of what fictional statements they can make ("I walk down the corridor and turn left at the T intersection"). They can also have agency with respect to the nature of the following state (maybe in some game a check can be made to see if I 'found a treasure', and if I invoke that and succeed then I have entered the 'found a treasure' state). Usually that kind of agency comes with 'strings attached', which 'drive the story'. Maybe in this hypothetical game failing 'Found a Treasure Check' means 'found a monster' state is entered. This is what we mean by 'staking something', the monster is dangerous, the player staked the character's existence on the check to find a treasure.@Crimson Longinus
What you and @FrogReaver are talking about seems more like a sense of ownership than a sense of agency to me. You are talking about feeling that you have complete control of what belongs to you. Agency is about action. Acting on the external world. It's about affecting change in the outside world rather than protecting what you already have.
So I do not think you need specific mechanics to have a sense of agency. What you do need is a shared expectation that everyone at the table is committed to respecting and following the shared fiction over their personal conception of the things they "own". Those expectations are very much a part of the rules of many games. When we can disregard fictional positioning anytime it is convenient than there is no real agency.
Thus it isn't totally unfair to say that there are "agency with respect to describing/taking an action" and "agency with respect to inventing a new game state". So the difference between 'classic RP', 'story teller RP', and 'narrative RP' involves some differences in the allocation of this to participants. In all forms of RP the players have the first type of agency WRT their PCs, with the caveat that there may be 'special game states' which suspend that, and that the conditions which must be met to transition from one state to another are not to be understood as 'voiding agency' but merely to 'giving it narrative force'. Without these conditions the RPG is not a game, because the 'game' part mostly involves those conditions, the problems that must be overcome by the characters. When we consider the Czege Principle then we find that primary authority over those conditions needs to rest with a party which doesn't face them, so game master (or maybe some rules process, such as in combat in D&D). The second type of agency is much less constrained, potentially, at least in game design terms. Instead it is constrained by the desired material content, theme, genre, etc which the participants wish to experience.
It is the second type that we have been mostly debating about here, except for this aside about "mental states", which if you analyze the model above you will see is not really important.