You have to be pretty relaxed to be good at Quake for example - otherwise you cannot get into "the zone" and think clearly (there IS actually room for thought in Quake).
Quake by its very nature amps you up. There's kind of like a magical point of the graph where your relaxation and your amped-upness meet; if I can find that point I'm in "the zone" become virtually unstoppable in-game.
I've found that Rez is designed to make that point relatively easy to find, which is exactly why it's the best game ever.
I've found the thing that helps me get into the zone with first-person-shooters, especially with CTF varieties, is knowing the map well enough to be able to predict where everyone is on the map based on where you last saw them -- how long a flag run takes, what the major routes are through the map, getting hit from that angle at that location means someone is sneaking around over there... and it doesn't even need to be exact, not "it takes 18 seconds to cap if you can run at full speed" (which is useful in its own right), but rather "the flag was picked up just as I respawned, and I'm here, and that's when a cap occurs..." or "I saw that guy when I died on that side of the map, and he should be crossing this area just as I respawn..." or "when I'm holding the flag here waiting for our flag to return, attacks usually come from that location, so let's pop a few grenades over there to keep it clear...".
Unfortunately, many FPS maps are created to be simple and _fair_ in the extreme, which ends up severely limiting the kinds of routes you can take (everyone's instinct is always to take the shortest/fastest one, or all routes are roughly the same length), or where the serious choke points are, so if you spend _merely_ weeks playing the same map, you can get a really good feel for the basic attributes.
In junior high, I used to play Counter-Strike competitively, and awareness (knowing the probable locations of opponents in any given scenario for any map) was the name of the game (apart from accuracy and overall competence). It was fascinating how teams would watch opponents' videos to pick up strategies and patterns much how high school football teams study scout tapes.
I remember playing Quakeworld. Before I quit, I wrote a strategy guide that elaborated on a "theory of qw". It was analogous to a well-known theory of poker. Quakeworld, with its unusual physics and excellently/accidentally-picked weapon attributes, convinced me that I could actually get smarter by competing in video games.
I really liked the little flourish in Okami where when you ran a trail of blooming flowers would follow your characters. And of course it was immensely satisfying to purify an area and watch the cut scene of all the evil being washed away.
Another game that fits this exact description, at least for me, is Tranquility (http://www.tqworld.com/). It's ten years older (and the basic graphics model shows it), but it's still actively maintained and looks pretty neat even on modern hardware.
A warning, though: unless you've been in microgravity before, or at least read the passages in Ender's Game dealing with reorienting "down" and altering rotational momentum to match targets, the controls might frustrate before they relax :)
Sort of like Pixeljunk Eden. Don't let the flowery decor, trippy music, and effortless videos of floating through space deceive you: it masks a brutally hard platformer which if my friends were not so committed to each other would probably have caused a divorce after failing to properly catch somebody at the top of the last level, which causes BOTH players to fall all the way to the bottom. And its a long way down, with plenty of time for bitter recriminations.
