Nekomata, all of the organizations you listed vie for power, wealth, and self-interest. They all are neck deep in MtG's black mana.
Not exactly. Seeking power isn't a trait that suits black alone, it is _how_ you seek power that determines if it is black or not. Green for instance wants power, but they seek to 'grow' that power, while blue seeks power by seeking to take control of it or 'create' it, and red doesn't so much seek power, as uses the power they do have.
Wealth isn't an inherently black trait either, it readily dabbles into other colors, particularly blue, in certain ways. The methodology for getting and reasons for desiring that wealth are the things that can make it black.
And I'd debate how self-interest applies there in terms of being black or not.
Why these aren't really that 'black':
Blue - this organization seeks the ability to influence in order to bring about changes, rather than for the power of that influence itself, they seek to solidify their power not just because they want to retain it, but because they want to minimize outside influences in their meddling to get greater control and see the results more clearly. They are also an organization that is friendly to those within it's structure, rather than viciously competing with each-other for status even within the organization, so they aren't entirely selfish. Their activities are closer to doing things like casting Control Magic or Redirect or using Clones and Body Doubles and sometimes causing incidental Gridlock sometimes than more black methods.
Red - They don't seek power for it's own sake, and only grab at it randomly and impulsively in simple manners when they need it for their purposes, and for the most part, are made up of people who already have power and are simply prone to misusing it in the sake of their ideals. Being violent and coercive are still within red pie space without having to be part black. They also aren't entirely selfish, they just have twisted ideals focused around artistry and beauty, so they may be just as likely to protect an artist or beautiful person as they are to bully an ugly plebeian. Black sees value wherever it can get power, regardless of how ugly that source of power is. This group despises ugly things and draws only the power they feel they need for their purposes or that pops up in front of them when they feel the desire for it under emotional impulses.
Green - black seeks power in any way possible, they wouldn't let themselves be bound by this organization's ancient traditions and rituals of ascension and status determination, black also wouldn't seek this organization's constant growth of membership, since they'd want to pare things down to control payouts, nor would they as stringently respond to insults at potential cost in the backlash when it isn't necessary, I'll admit they do have some more black elements than the other ones with the 'any means necessary' I mentioned for how they seek new businesses and that was probably a mistake I made there, and a better green design would probably be that they seek it within certain tried and true (if often illegal) methods, and not at the cost of their own people or other businesses.
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(tyranny, utilitarianism, censorship, lasers) or 
(selfishness, ruthlessness, dishonesty, shadows).