Okay since I'm not sure anyone has exactly compiled it yet, I want to give a quick breakdown of some of the control mechanics and how it ties in with the stats you can level up, since there seems to be a lot of general confusion. I'm just a player so it's possible this isn't 100% accurate...but I think it's right, given how I've been playing so far and also going off of this game's many correlations to Mario Golf among others.
Power: Simple enough, this determines how far your ball can potentially fly when you hit it. Adding more into this stat will occasionally lower some of the others, but is necessary to get your scores on the course lower, so it helps to keep an eye on regularly raising this while "managing" the others to more suit your particular play-style and keep them from falling too far into the negative.
Purity: This stat determines whether you have a natural draw or fade on your shots a.k.a. whether your shot-line has any sort of a curve to it while you're just aiming around and so forth. Going lower or higher in this stat will increase the curve to that direction, while keeping this stat centered will keep your shots straight.
Strike: Since this is being measured in accuracy, my guess is that this stat determines the potential deviation your shot suffers from when you don't hit your alignment perfectly center at the end of your shot (aka the third button press of the "three-click" golf swing) Higher ability means less penalty, and thus a larger margin for error while still getting the ball to generally go where you're aiming it.
Ability: Again slightly speculative, but I believe this stat determines the degree to which you can shape your ball flight by adjusting your hit point (this is what you do when you press X and change where the little dot is aligned on your golf ball). Low hit point means the ball will take a higher arc and thus also land at a sharper angle, high hit point means the ball will travel lower and be more prone to rolling forward after hitting the ground, while adjusting the hit-point left and right will bend your flight arc a little bit, which can be useful for hitting around obstacles or slightly changing your approach trajectory. So, higher ability score means you have more direct control over your flight-line.
Spin: This determines the severity of topspin and backspin when it's applied on the ball (this is what you do using L or R while you're setting up your shot and it puts a little highlight over the ball indicator). This takes effect once your ball lands and can cause it to roll forward further in the case of topspin, or stop short or even roll back a little bit with backspin. When you want to use each depends on the situation but strong spins especially in conjunction with adjusting your hit-point gives you a lot of options in how you want to approach a hole.
Beyond that, you get ZL and ZR, which are used to help set up chip and approach shots. It's worth mentioning that you have three different approach ranges to work with depending on which buttons you do or don't have pressed. ZL is the shortest range, both simultaneously is in the middle, and ZR is closest to your full-power hit. The exception to this is putting, where holding ZL puts you into a short-range putting power scale while ZR gives you long-range. Pressing Y puts you into "precision mode," letting you use up/down on the left stick to create a kind of shadow-indicator that doubles on your power bar to help you work out exactly where on the power meter you want to stop to get the ball where you want. Swinging all out isn't always the best option!
There's also normal vs. "focus" mode....but I haven't toyed around with that yet enough to figure out what it does. Hopefully the rest of this helps, though!