It seems to me that a very general case for "finite memory games" is the case of a "non-deterministic finite memory game with $n$ players". I made this classification many years ago (about 5 or 6) mostly out of concern of conceptual (not mathematical) classification. And I hope people see the answer in that light. The advantage of this formulation is that it can be quickly be specialised to more specific cases as the need arises. The version I describe here is not fully general (for the sake of brevity), but general enough that one can easily see how to make it more general if need arises. At the same time, I will admit complete ignorance of usage of this term in economics, combinatorics, mathematics etc.
Below we consider a description of a "non-deterministic finite memory game with a single player". Some further assumptions are (that can be relaxed if required): (i) From every state there is a path to both "Win" and "Lose" state (should become quite clear soon). This condition is probably a little restrictive (especially when we might have more than one player), but makes the description below fairly short. Shouldn't be difficult to relax this condition. (ii) In principle, the game is of perfect information. That is, at least in principle, the player can always figure out which state of the game is the player currently occupying.
Making a generalisation to accommodate for both (i) and (ii) isn't that difficult but I have tried to keep things simple (so the basic idea can be explained in a not too long answer). Now here is the description:
(a) We have a finite set of states denote by set $S$. The game always includes a "Win" and "Lose" states. If there are $n$ number of (main) states, then we have $M=\{s_1,\,s_2,\, s_3,....,\, s_n\}$ (main states only) and $S=\{s_1,\,s_2,\, s_3,....,\, s_n,\, Win ,\, Lose \}$. The state $s_1$ could possibly be thought of a start state.
(b) We have a finite number of actions for the player. So if there are $m$ number of actions, we can denote them as: $A=\{a_1,\, a_2, \, a_3, \,...., a_m \}$
(c) From each (main) state $s_i$ and for each action, there are finite (but potentially more than 1) arrows that lead to other states (possibly including $s_i$ itself). Hence we can think of a transition function from $M \times A$ to subsets of $S$.
An example w.r.t. (b). If we think of a computer game being played with a controller with just two buttons A and B for example, then the set of actions $A$ can naturally be thought of as $A=\{doNothing,\, pressA,\,pressB,\, pressAandB \}$. At the end of each clock cycle, the game is updated according to the action of the player.
An example w.r.t. (c). In a single player game of a traditional computer version of tetris (that's the most natural example I can think of), the randomiser that hands the pieces/tetrominoes can be thought of as adding the non-determinism (in a natural way). Of course, this might differ a bit from the actual specific implementation, but we are talking about a conceptual analysis.
Now, as far as (elementary) mathematical analysis is concerned (without accounting for any time-complexity concerns), each state in the set $M$ of main states can be marked as $W$ or $WL$. A given state $s_i$ being marked $W$ means that the player can "always" win the game from that state if he plays well-enough (winning the game means reaching the "Win" state). A given state $s_i$ being marked $WL$ means that no-matter how well the player plays from this point on, a win can't be guaranteed. Though while writing this it occurred to me a significant concern could also be whether it might be possible (just by chance) to be promoted to a $W$ marked state (from a state marked $WL$).
Similarly each individual transition in the transition function (described in (c)) can also be marked. Each transition can be marked as one of $w$, $wl$, $l$. A transition being marked $w$ means it only leads to main states marked $W$ or to the "Win" state (one can suitably account for arrows from a given state to itself). A transition being marked $wl$ means that it leads to at least one state marked as $WL$. A transition being marked $l$ means that it only leads to the "Lose" state.
One can (using the previous two paragraphs) devise a general naive algorithm (ignoring computational complexity) that can mark all the states and transitions in any "non-deterministic finite memory game with a single player" (and hence a perfect strategy too ... if one exists from the start state).
[EDIT2:] However, I was viewing this thread again and it seems to me that there is one further distinction that is worth making (and shows how many distinctions can be made even in this simplest case). Suppose we run the algorithm for the above mentioned game and the algorithm tells us that the start state is $WL$ (but not $W$). Hence we will know from that we have no definitive way of winning (with 100% chance). So is there something else we can add? I think there is atleast one point that can be added.
Suppose the player hates to lose (and if there is any risk of losing, the player would want the game to go on forever). Now the player would want to know whether the game could be made to go on forever with some strategy. It seems that this can be handled too. However, notice there is one further slightly subtle aspect still. We know that the start state is marked $WL$. However, this doesn't tell us (by itself) that we might have few "chances" of being promoted to a state marked $W$ at some points! But once again we would want to know that in case those "chances" all go against the player (player isn't promoted), whether after that the player would still be able to force the game to go on forever! Because ideally, if the game could be made to go on forever after taking all the chances (of being promoted), that's what the player should be doing.
This probably needs to be worked a bit further to make it more rigorous, but I am reasonably confident that the basic idea in above paragraphs is correct. [END]
To see, how flexible this definition is (and how it can be easily accommodated for most variations in "finite memory games"), here are few examples:
(1) [EDIT2] My example w.r.t. to "chess" and "go" was described rather badly (and hence this re-wording). Let's first think of entirely different examples first. A reasonable analogy for generalisation of the above single player game could perhaps be written as "simultaneous non-determinisitic game of two players". One might think of a computer game like "Frozen Synapse"(https://en.wikipedia.org/wiki/Frozen_Synapse). The details are not too important. Another example, could be suppose a board-like game (not too different from checkers or chess) being played online by two different players from different locations. None of the players can see the other's board, but both move there pieces in the same turn and click the finalise button after decision. The game turn proceeds after both players have clicked on finalise. The "non-deterministic" qualification means that we are allowing for some possible "roll of dice" (even for a specific combination of both players' action).
Now let's think a bit about how we can modify the above scheme for a game like chess for example (which is "turn-based" as opposed to "simultaneous" .... and well involves no roll of dice either). We can just colour all the game states as either "white" or "black" (the start state would be coloured "white" indicating that it makes the first move). Let's think about a bit for the set $A_1$ or $A_2$ (actions for "white" and "black" respectively). For example, the set $A_1$ can be made to correspond to all possible moves of individual pieces (and hence $A_1$ will be finite). Similarly for the set $A_2$.
If a state is coloured white (meaning white's turn) then any action from that state is an element of $A_1 \times A_2$. Let $a_1 \in A_1$ and similarly let $a_2, \, b_2 \in A_2$. Observe that on a state coloured white both $(a_1,a_2)$ and $(a_1,b_2)$ will lead to the same state. Furthermore, after an action is selected the next state will be one coloured black.
Finally we do need to account for the fact that once a piece (say a "white piece") is thrown out of the board, the elements corresponding to it in $A_1$ lost their "meaning" (and perhaps can be put equivalent as default to some other action). But I guess that's really an artifact of the way "domain" of "transition function" is defined. Maybe that can be handled in a more natural way?
The summary of that simply is that for two or more players, the "simulaneous" version is general enough that it can "include" the "turn-based" somewhat naturally (though with some artifacts). [END]
(2) Consider a computer version of tetris with the feature of endless play included. We can simply modify our above definition to "remove" the "Win" state entirely and just keep the "Lose" state. Now instead of each main state being marked as "W" or "WL", it is only marked as "O" (guaranteed possible to orbit or play forever with perfect play) or "OL" (not guaranteed possibility to play forever with any strategy .... but with enough luck on our side possible to reach a state marked "O") or "L" (guaranteed to lose eventually from this point).
(3) Sometimes computer games can also have bugs which can put player in an endless situation --- when the actual goal was just to "Win"(reach the end) or "Lose"(gameover screen). As far as I know, this was actually the case in a rather well-known genesis game.
To accommodate this kind of possibility we admit the possibility that from a given state there may be no path to either the "Win" state or the "Lose" state (meaning you are stuck forever without any possibility of winning or losing). The classification of main states will also increase from just $W$ and $WL$.
(4) Finally there is the issue of perfect information. Fog of war in RTS game is rather conventional example. But for the sake of our case, consider a board game with "fog of war". In that case the player can't know the complete state of the game with certainty (because he can't view all of the board). These kind of issues can also be handled with the some modification to basic definition described above (possibly by making a distinction between "player state" and "game state"). I think the player state could then be idealised as a finite set of game states. But I haven't thought about this situation that much.
(5) I haven't thought that much about probabilistic concerns (each arrow in a transition marked with a probability), but they might probably significantly complicate the maths.
Obviously this answer doesn't address the (more) mathematical aspects that are specific to board games ..... and neither the computation complexity concerns (that people might have for practical reasons). In some sense, I would just note briefly though that a very similar question can be raised also for computer games (possibly mathematically more useful to pose for more specific genre) .... as has been raised for board games here .... once one regards them as finite memory games.
Edit: Made number of corrections on various points.