Style VS. "Correct Moves"

Whenever I read chess books or use the Chess Mentor on this site, I notice they always have one "correct" way to go about things. I suppose I agree with this because based any position, the way to get the upper-hand will be either a tactical combination or the fastest plan to take advantage of a positional weakness. But, with that in mind, how can we have style when the best solution is what the position requires? 

Always be careful of the word "best" if it's not defined. The word "best" always depends on what your values are. Nowadays opening books, game databases, lists of gambits, and chess engines usually list moves in descending order by their popularity, estimated strength, estimated soundness, etc., so that there isn't a single alternative presented. I've also found that what is "best" for a high-rated player is different than what is "best" for a low-rated player: high-rated players usually look for complicated, tactical positions where they can outcalculate their opponents, whereas low-rated players look for simple positions where they won't get outcalculated. Therefore where a high-rated player would say 1...c5 is the best response to 1. e4 since the game then becomes complicated and tactical, a low-rated player would say 1...e5 is the best response since the game then becomes simple and positional.

Therefore I assume "style" comes from some background philosophy that often chooses moves that aren't the top recommended moves, or (as in the case of Tal) chooses moves whose outcomes aren't yet known.

Some positions have only 1 solution or type of solution.  Basically a player shows their style in those positions where there are options.

I was just going over a game where Kasparov, who was of course known for his brilliant dynamic middlegame play, traded straight into a favorable endgame, and won with technical play.  Why?  Because that was the best way to win that position and he knew how to do it.

In other positions there are options.  I don't have one off hand, sorry, but basically something like a middlegame with slightly better pieces and some space on one side of the board (middlegame chances) where you also have the opportunity to trade into an endgame while maintaining your better pieces.  If you analyzed that both give about the same opportunity to win, you'd show your style by choosing what you like best.

So basically I can take any "move and win" type puzzle with a grain of salt because even though the "correct" move might win material, other options could lead into an equally favorable positional advantage. 

Well, yes, that's true! :)

Although it's very rare that an alternative will give an equally favorable advantage.  Usually if there is an alternate way to win, it's more difficult at best and practically impossible at worst.  Particularly in tactical puzzles it's worth it to find out why the wrong moves are wrong.  If you really thought your move worked, stick with it and figure out why it wasn't best.  Tactical puzzles are checked with engines now-a-days, so solutions are reliable.

Even so, in those puzzles where there are alternate solutions, the puzzle's idea is very important to learn!  In future games it may be the only way to win.

shell_knights explanation is good. In some positions, there is a clear best move, while in others, there are 2 or 3 good moves, and it is a  matter of style on what option tht player chooses

Now you're talking about something different than I thought you were talking about. If you're talking about a *tactical* sequence of moves like in a puzzle, then of course there is almost always one best answer and that answer is actually measurable by least number of moves to an overwhelming position. I thought you were talking about *strategic* moves that hadn't yet gelled into a tactical situation. In tactical situations it is simply a matter of calcuation, so style doesn't enter into it.

I think that the most succinct example of style versus correctness was the famous game where Tony Miles beat Karpov. Miles was black and played against 1. e4       a6!? and a ..b5 system. Miles won. Karpov said afterwards that Miles' position was worse for the opening. But some GMs thought after the game that Karpov had a weakness against oddball openings that were objectively pretty dubious!! Miles was an interesting player. In unorthodox chess openings Schiller says that Miles may have struck the right balance in his GM chess career of being unorthodox without being a total lunatic! That said, Miles didn't play..a6 again against another GM as far as I know. I read somewhere that Tony Miles played Sicllian Dragons instead which was probably more sensible. 

I remember Kasparov talking about Karpov's playing style. Kasparov described Karpov as being a bit like a creepy spider that was on you and wouldn't let go.