Building off my previous three-axis model, some analyses of various combinations thereof.
To recap, three axes: Pragmatic vs Heroic (very roughly, "fighting dirty" vs "playing fair"), Strategic vs Tactical (very roughly, "logistics and campaigning" vs "maneuvers and synergies"), and Narrative vs Challenge (very roughly, "revealing/discovering lore and story via combat" vs "overcoming difficult obstacles via combat.") All of these are only "axes" in a loose sense, in that a game can feature stuff from both sides; instead, think of it more as estimating the rough average of a given game, so one that has some strategic elements but lots of tactical elements would be moderately tactical.
Pragmatic Strategic Challenge: The classical mode of D&D, what is often referred to as "Gygaxian" or "player skill" play. Gameplay is structured in such a way as to present difficult obstacles, some perilously so, which the players must overcome with wit, guile, and resourcefulness. It is precision-designed to expect out-of-the-box thinking and comprehensive preparation on the players' parts. Creative thinking in this mode centers on (more or less) finding ways to metaphorically "break the rules," meaning the rules of engagement, because if one brazenly attacks head-on, defeat is essentially guaranteed in many cases. Instead, the players are rewarded more (or punished less) for subverting the enemy's preparations, sneaking through the Great Wall rather than trying to pierce it or destroy it.
However, this style risks becoming stale if play becomes too consistent, what I refer to as the "standard operating procedures" problem. Excessively defaulting to well-made SOPs removes several layers of the challenge. This is where players can "optimize the fun out" by the perfectly-valid search for reliable, effective countermeasures to expected problems and powerful, general tools/strategies/behaviors that, even if they don't guarantee success, certainly make it way more likely that one will succeed. (In effect, developing the technology of strategic theory within the confines of the game.) Sadly, a common response to the SOP problem is DM-player arms races, which mostly just serve to frustrate both sides. A better answer is usually to, as Gygax did, create entirely new incentive structures (the "domain play" transition) so that the old SOPs are still technically useful in their domain, that domain is just revealed to be more limited than expected.
Heroic Tactical Challenge: The mode flirted with by 3e and fully committed to in 4e. Player characters are basically fantasy action protagonists, Big Damn Heroes to use the trope term. The focus is on the tension and excitement of round-by-round conflict, where a smart move or a foolish blunder can turn the tide of a critical battle. Gameplay is structured so as to reward teamwork, situational awareness, exploitation of interlocking gameplay elements, spatial awareness/terrain exploitation, and tempo. Creative thinking focuses on predicting the actions the enemy will take, setting up useful combinations of effects, and altering the round-by-round incentives for both enemies and allies alike. Instead of modifying the rules of engagement, one modifies the field of engagement, or the participants therein.
Of course, the problems here are obvious and well-trodden, but in the interest of fairness, I will lay them out. Firstly, a failure to actually provide the required texture: combats on empty flat planes or uniform straight hallways or the like. This flattens the experience, leaving little to do but drop your strongest effects, repeating until the fight ends. Secondly, a failure to offer variety. Because centerpiece of this experience is the enemies themselves, especially the actions they take and their choices and incentives, it is critical to keep them feeling fresh. This requires a great degree of creativity on the DM's part, and an ability to see how mechanics can interact with each other. Finally, there is the dreaded "treadmill" problem: if the characters grow stronger and their opposition grows stronger, it is easy for things to feel like nothing has actually changed, it's just "Numbers Go Up" gameplay. Unfortunately, most of the fixes for this involve at least some injection of narrative considerations (e.g. featuring identical or near-identical enemy stat blocks over the course of a broad swathe of player levels, so the players can feel powerless early on and powerful later, after they've grown.)
Pragmatic Tactical Narrative: An unusual combination, one I'm not sure is used all that much--though I welcome examples or suggestions. (Perhaps Numenera?) I would expect this to emphasize survival, perhaps even horror. The big picture is insoluble in Lovecraftian/cosmic-horror cases, so there's no point to "strategic" play, but little picture goals can be quite achievable if one can make the savvy moment by moment moves, meaning that correct responses to individual events/stimuli remains significant. And these conflicts further the purpose of examining or illustrating the world. If we flip from Pragmatic to Heroic, then the option of something like supers gameplay comes into focus here. Alternatively, perhaps something gritty like noir, post-apoc, or cyberpunk/tech-dystopia would work: fighting individual fights and focusing on surviving this battle, this mission, with relatively little bleedover from one mission to the next.
I can see at least a main difficulty here being the question of how to maintain interest and tension within and between individual combats. Something like Shadowrun achieves this through set dressing, the whole "near-future" concept and sci-fi/fantasy mashup enabling something unexpected to pop out of the shadows or appear just around the corner. A second issue (common to all Narrative options) is that whatever theme or concept is described in, explored by, or discovered through combat has to be actually interesting in order to warrant description, exploration, or discovery. This puts heavy pressure on the GM in a completely different sense from the previous style: instead of needing creativity in mechanic design, the GM needs creativity in storytelling in order for the experience to feel rewarding.
Heroic Strategic Challenge: Exact inversion of the previous, and one I would call even more unusual--possibly even not used, since I could at least come up with some kinda-sorta examples after thinking on the previous. I suspect this would be something like a more positive spin on Dark Sun: starting off with a world that is an absolute $#!%hole you can barely survive in, with the long-running challenge of "build from hell your paradise." The hope is real (it is heroic, after all), but difficult (hence "strategic challenge.") You have to think long-term, building up reputation and infrastructure, overcoming challenges and mitigating new or unexpected complications.
Frankly, the biggest challenge here just seems to be fitting all these pieces together. I'd honestly really love to hear of anything people think might count as "heroic strategic challenge," because as it stands, I'm not sure what would qualify! Having an actual example game to pore over and examine would do wonders for finding places where it's liable to run into issues.
Obviously with 3 axes, we have 2^3 = 8 different combinations, but I'm not super interested in breaking them all down. This was more a test run to see how it handled both extant things and hypothetical ones.
