TN, you nailed my complaint about chess styles on the head.  We all have elements of any style that is worth mentioning.  The distinctions are usually artificial and predetermined. Most attempts at "chess styles" tend to favor one way of playing over others. This seems to be a more sophisticated division than most, but I suspect it still has the two fundamental flaws that I just described.

Let's see... judging from my handle.  I can't tell.

I have always been fascinated by discussion of chess styles. Most chess players would describe themselves as either tactical or positional players, but this characterization is too restrictive . 

Personally, I like the approach used by Lars Bo Hansen in his series of books. He uses a quadrant with two continuums:   logic vs intuition and fact(calculation) vs. general concepts. From this he defines four principal styles - reflectors, theorists, pragmatics and activists. Importantly, however,, he acknowledge that one's style can often be a combination of two styles. depending on at what point you lie along the two continuums. Based on his definitions, I am principally a reflector. 

Recently, Everyman Chess launched a  new series of annotated game collections called "Chess Secrets" So far they have released Heroes of Power Play, Giants of Strategy, Giants of Power Play and Great Attackers.

They have defined classically direct styles as follows: 
" a universality of play that embraces all styles, in a sense, and transcends narrow boundaries – a style that is based on such overarching attributes as clarity, energy,toughness, ambition and a fundamental sense of analytic “correctness”." 

Players they have defined as classcially direct are Carlsen, Anand, Fischer, Smyslov and Rubinstein.

They have defined power play styles as follows:"the power play style can be summed up in three words as a blend of preparation, psychology and dynamism. Here is a list of specific features:1) Playing aggressively with Black as well as White. This requires the acceptance of risk.  2) A big attempt is made to win the theoretical battle by surprising the opponent
with a new move or a new way of handling a sharp opening system.3) The stakes are set high, with the
enemy king often being the target, rather than a weak pawn or square. A pawn sacrifice is often used to accelerate the attacking process. 4) Threats to the king are combined with threats to another piece or pieces in order to overstretch the defence. 5) Rather than a drive towards clarity, there is an active search for tactical blows hidden at the end of variations.
Whereas strategical play is a step by step process, power play might be divided into two distinct parts:
6) The first is a dynamic strategy that, if it outwits the opponent, creates situations where combinations are likely;7) The second is the use of tactics to exploit the edge that has been gained. 8) Playing the man as well as the board. For example, an opponent whodislikes messy tactics is forced into a double‐edged fight, even if objectively they should turn out well for him. 9) Time trouble isn’t regarded as a nuisance but rather as the chance to confront the opponent with problems
he will struggle to solve. 10) Above all, the byword of power play is imbalance. A position with mutual
weaknesses and a disrupted pawn structure contains room for creativity – and blunders by the opponent."

They have defined the following players as power players: Topalov, Geller, Bronstein, Alekhine and Morphy

I could not find an pdf excerpts for the Great Attackers or Giants of Strategy, but the definitions of these two are probably closer to our traditional understanding. Their list of Great Attackers is Kasparov, Tal and Stein and the list of Giants of Strategy is Kramnik, Karpov, Petrosian, Capablanca and Nimzowitsch.