The term avatar, if it is to be deployed usefully, refers to the means by which the player of a game interacts with its fictional world. But in which case, the avatar as a representation is secondary to the avatar per se – surely the parser was the source of interaction in Colossal Cave Adventure, which after all did not represent the player at all. If we presume (following game design as make-believe) that the avatar is the prop which prescribes that the player imagines they are in the fictional world of the game, it is the verbal representation of the adventure game that achieves this by virtue of the use of second person narration. The MUD shows this is not just an idle observation: MUD players certainly had avatars, even without any explicit representation.
But if there need be no graphical representation for the avatar, then there is a distinction between the avatar as the source of interaction and the avatar as representation. The latter is merely a doll that prescribes we imagine how we appear within the fictional world, not that we are present in the fictional world. (To put this another way: the character sheet in a tabletop role-playing game need not have a picture or a written description to function as the player's avatar).Thus most of the things that are conventionally called avatars are merely dolls representing the avatar, what we might term an avatar doll.
Consider the alternative: that what we mean by avatar is precisely the prop that specifies our appearance in the fictional world. Then MUD players only have avatars if they type a description of their character (or if the name alone were considered to represent them in that world). This is tolerable, but it shows that our focus has become what we communicate to other players about our appearance in the world, which would be meaningless in a single player game (although a case could be made that players can communicate representative aspects to themselves, or that game makers can do this to their players). We could enter the fictional world of a MUD without a name or a description, so the avatar cannot be either of these things. It therefore cannot be representational at all.
Also consider that the first person shooter, in its classic form, shows only a gun and perhaps an arm – is the arm our avatar? Or the gun? We might prefer to recognize that the gun implies our ability to interact with the fictional world and thus deserves to be called the avatar, but this is a case where there is little or no doll to be found. Indeed, this distinguishes first person perspective from third person in most cases, although the first person computer RPG often still shows the doll on an inventory screen. Overall, the doll seems secondary to the avatar as such when we consider games presented in a first person perspective.
The situation is similar with the racing game. In first person, the dashboard reinforces our presence in the fictional world, but even without this we would still be driving in that world; the dashboard is closer in function to a doll than to an avatar. Similarly in third person: the car is the equivalent of a doll (or perhaps more precisely, a model our doll is presumed to be driving), but we would still have the avatar without the car model, since in first person any explicit representation is entirely optional.
Thus we can conclude that the doll (or model) in any game is a prop that prescribes we imagine the details of our presence in the fictional world, whereas the avatar is our capacity to act within that world. Crucially, this suggests the avatar is not representational; it is wholly functional. And this transforms our expectation of what the avatar must be completely. Given this, nothing that serves a representative function deserves the term 'avatar', and whatever does so may better be considered a 'doll' or 'avatar doll'. The doll implies the avatar, but the avatar is in no way dependent upon a doll. We act in fictional worlds even without a doll, and thus the avatar must not be a representational prop at all.