I guess either. From what I hear if a pitcher has a low BABIP his defense is credited for making a lot of plays behind him. What if most of those are just routine plays? Certain pitchers are looking to force a ground out or fly out instead of getting a strikeout. It seems the defense is getting more credit for catching a routine fly out rather than the pitcher for causing it.
As for batters why not just look at batting average instead of BABIP? An out is an out. If you ground out or strike out it's the same thing (unless you're trying to advance a runner). Maybe I'm not understanding this stat correctly.
The first thing to note about BABIP is that it shows reasonably strong predictive capability - that is, without even understanding what it is, precisely, we can work from the fact that it generally (not always) has predictive value about whether a pitcher or hitter is due to regress. So, let's say a batter has an unusually high BABIP, like say .330 - .300 is average -
and his career line isn't unusually deviant - because a player can influence his own BABIP to an extent. You probably see that high BABIP reflected in deviant average, OBP, slugging, etc., etc. So you could generally expect that those numbers will go down. They won't always, but usually they do.
So, then getting into the meat of BABIP, it basically comes down to three factors - defense, luck, and and changes in talent level. So, those three factors cause a player's BABIP to deviate from .300. Defense and luck are out of his hands. Are you hitting a hard liner to Miguel Cabrera or Evan Longoria? One of those guys is obviously more likely to come up with it. Or you just get unlucky. A guy makes a highlight reel play against you. But then talent level does play a role. Look at Matt Kemp, for example. He's always posted an incredibly deviant BABIP, about .350 for his career. He's not just constantly going up against shitty defense and getting super lucky. Over five seasons, that stuff evens itself out. What's happening for Kemp? Hard contact is usually the answer - the more solid contact a player makes on the ball, the better his BABIP. Kemp is also pretty fast, as it goes, which influences his BABIP as well. And through the first month of the season Kemp has flirted with a BABIP of .500. Is it all luck? All poor D? I'd imagine it's influencing it, but 11 home runs also indicate that he's making fine contact. So, on the batter's end, they're able to influence BABIP a lot, which is why most of the game's best hitters have deviant BABIPs.
On the pitching end, the same factors come into play. I'll use an example of a play I just saw - Aaron Cunningham made a beautiful, somewhat lucky catch in right just by reaching out his arm after losing the ball in the sun. By rights, that was probably a hit. Now, to answer your question about inducing grounders or flyballs - that's factored in. BABIP takes account of pitchers ensuring that their batters make poor contact. As a result, you see guys with high GB% post low BABIPs. The same goes for guys producing easy flies. BABIP is calculated a lot more subtly than it appears. Strikeouts, however, are given more weight, so high strikeout pitchers usually have the best BABIPs - Verlander posts a pretty deviant one every year, for example. I checked Nolan Ryan's and he has a .265 career BABIP, so you see how that works. I checked Derek Lowe's, he was the first groundball pitcher I thought of, and when he was in his better days he was about a .280 BABIP guy. So, the answer to defense getting more credit for the play than the pitcher isn't entirely true - BABIP rewards pitchers who pitch for weak contact. But, a pitcher that surrenders hard contact that is made up for by excellent defense or luck will see their BABIP rise.
As for why not look at average for a hitter - average sucks. It just really, really sucks. It tells you so little about a batter. But BABIP isn't the answer to evaluating the talent level of a batter - there's better answers to that. While BABIP
can reflect something about a hitter's talent, it doesn't make it clear that it does. If you're interested in evaluating a player's talent level, wOBA and WRC+ are better evaluators of that. But I refuse to launch into a diatribe about those

.
Batting Average on Balls In Play is one of these new things, and it warrants a bit of merit. It kinda shows how a pitcher pitches to defense.. I've never really heard batters using BABIP.
It's not like arbitrary like WAR (of which there is no real equation or easy way to describe who is better then who, it's just a number of which no one can explain how it is determined).
Wherever did you get this idea? There's a very simple formula for WAR. Fangraphs explains it here:
Link
Take wRAA, UBR, and UZR - for hitting, baserunning, and defense - add them together, adjust by position (third base is harder than first, etc.), adjust to replacement level, because the above numbers are based on league average, and convert runs to wins - 10 runs = 1 win. Pitchers are done almost the same, except with FIP instead of the three above stats. That's a pretty skeletal explanation - if you want in depth rationale about where each of the individual numbers come from, why they're used, why ten runs equals a win, etc., etc., you can peruse the Fangraphs link, which contains in it a link to a very in depth explanation of each component of WAR.
The point is that it's the exact opposite of arbitrary - it's extremely well documented on how to get WAR, and justification is made for every step.
EDIT: I should note that my above skeletal explanation of WAR is for fWAR, not rWAR, which is calculated slightly differently.