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Starting with the second: 

These categories are not only not exhaustive, they're...   Hrm.  Not quite illusory, but it's like pointing at clusters on a scatter plot?  It's clear there's something going on there, but if a given person goes "Oh, that cluster has a lot of subtypes", they're super-correct.  These clusters are made up of subtypes all the way down to the level of the individual experience.  If that works for you.

Now, to the first:

This is a messy question!   So, for starters, not all games actually imply or support a singular playstyle, or support the one they claim.  Ron Edwards of the Forge spilled a stunning amount of digital ink on the fact that original Vampire: The Masquerade promised something collaboratively narrative, but didn't actually support it for shit. 

I'm going to set aside the phrase "reward system", because it can also mean really specific devices like Fan Mail and Bennies and Xp, for a second.  I'll come back to it after this paragraph...

A game that's working well for sure encourages you to do what it wants by holding out specific kinds of fun, by encouraging you or leading you to visualize a playstyle, to imagine the payoff - solving the tactical puzzle, the beautiful pain in affective play, and so on.  Ideally, it highlights this by putting the devices for that playstyle right up in your face, in the form of stats and dice and cards and other stuff to grab and engage.  Now, not all players are necessarily receptive to whatever that is (and GMs might not care about that either), plus games aren't generally plain about exactly what they're holding out (having often arrived at it by trial and error or blind inspiration in design), so you get a lot of games being overwritten on this count.  Which is mostly fine, if the group finds their fun; it's just crap when the group wants to engage the design and misses because it's unclear.

...So, by that reckoning, most of a game is a big-ass reward system set up to deliver the goods on the intended types of fun that make up the envisioned playstyle, and calling the "last mile" or mechanically visible delivery device the reward system is kind of odd.  Especially when those "rewards", being things like Xp, etc, aren't actually the rewards - they're just the shiny bait that draws you in to the machinery.  The fun is the reward.