I'm thinking specifically of flow as a purpose*. As a player may seek to define skill and display it, they may seek to find flow, experience and evince it. We could say that skill can show up in any game, so I am not confident that
purposes is being parsimoniously adhered to.
As
@EzekielRaiden and others have helpfully noted, I keep getting tripped up as to what each model is addressing. So I would love to see a cleaner definition of "purposes" so as to be able to confidently exclude say flow from among them, and include metrics of grading success... which on surface seem to me on the same layer as tempo. Perhaps the answer is found in the binary, but I would rather let
@EzekielRaiden speak to that.
[EDIT *So by way of example, the binary would be
Tempo / Flow.]
Well, as before, I struggle to see how this differs from "make a well-designed game, regardless of what it's for." Analogically, "what about a person who just wants their vehicle to drive smoothly?" Smooth operation is an ideal of design, effortlessness. That is, certainly, a thing players will want to pursue, but I'm not sure it's a purpose for which a game can be made. They instead strike me as a positive quality a game should evince if it is, in fact, designed well for whatever purpose it has, when the desires of the players are in sync with that design. Again for an analogy on why that last "in sync" bit is necessary, it doesn't matter how flawlessly the stick-shift works in a well-oiled-machine sports car if the driver
wants a pickup to haul things; the sports car won't fulfill that want, no matter how it "flows."
Before I actually respond to your request to pick apart the difference here, though, there's a critical concern in your comparison I want to point out. You are speaking of
the player as the one who "seek[s ] to define skill," but in general that's not really how S&A games work, at least as I've understood it. The
system (or, perhaps, the DM, but usually the system) defines what "skill" is--what it means to succeed at a task, and moreover, the degree to which one succeeds at it--and the player is generally not involved in that process at all. The player often is involved in selecting
which tasks to attempt (e.g., which attacks or skills to use in D&D), but things like target numbers, probabilities of success, what tools are
possible for manipulating those probabilities, etc. are all fixed within the game. A player
inventing new rules for what qualifies as "success" is generally frowned upon in Score-and-Achievement contexts, because as a rule that is extremely sensitive to abuse. Stereotypical playground "yeah well I have an INFINITY PLUS ONE sword!!" stuff.
Coming back to this: What does it mean for a game to be designed for "flow"? Whatever it is, I'm certain it is of aesthetic value (the word fairly drips with such meaning) but in terms of something the player's actions can drive toward, what
is "flow"? From a superficial reading of the word, it just sounds like "the game does what it's supposed to do very well." That, to me, doesn't sound like a game-purpose; it also doesn't sound like something players can
pursue so much as
witness. It sounds like a word for "a game that has good design."
"Tempo," on the other hand, sounds like a word for
pacing, that is, the rate or sequencing of experiences. Obviously, to some extent, this depends on how the game itself is used, but since you didn't specifically give much definition for "tempo" it's hard to dig deeper on this front. Hence, same question as above: what
is "tempo"? Like, how does one design it? Is it possible to have a game where tempo is the only thing play is designed for? Can a game completely ignore tempo and still be a good, effective experience for its players?