I've always assumed I run pretty straight forward black trenchcoat games. That's what I like. It is, I feel, what the OG Shadowrun was intended for. I also extend that definition beyond the players to how it describes the world - as a mean, and dangerous, and unpleasant place. Like Marcus said earlier in the thread. Maybe you're professional, maybe you take the job seriously, and maybe Mr Johnson still screws you and there's nothing you can do about it because even in the shadows the corps own you.
And then, in my last session, the magician decided to try and shotcut a run by summing an air spirit and asking it to steal some high grade cyberware from a clinic. It didn't work. It not only failed, it failed embarassingly, it high profile failed, and this unfortunate air spirit was caught on EvoTube desparately trying to escape this clinic with this cyberware. When the laughter stopped I had to ask "Who brought all this pink hair dye into my game?"
Reflecting afterwards about the characters in my game, some more than others, and about how I present the idea of Shadowrun to my players, I've begun to think that I don't really run a straight forward black trenchcoat game. One of my players suggested "Black trenchcoat with pink lining", but I think pink trenchcoat works just as well.
My players are professional and highly skilled criminals who take their work serious and aim to be a well oiled machine of espionage and occasional murder. They make it clear in the meet and on the job that they're here to do business. But they're also all a little eccentric, a little broken, and a little mad. But I always say you've got to be a little crazy to be a Shadowrunner. No truly sane person would do it, not when being a wageslave or a BTL junkie or even a go-ganger is easier and safer. There's other ways to escape the soul grinding mundainity and corporate oppression of the sixth world.
I'd never heard of black mohawk or pink trenchcoat before this thread came up, but this is how I define pink trenchcoat, as, I think, the best description of my shadowrun games: A game set in a sleazy gritty cyberpunk dystopia, where the PCs are highly skilled professional criminals working high risk/high reward jobs, because they're too damned crazy to do anything else; they're eccentric, distinct, and stylistic characters as much defined by their personal foibles and madness as by their grim determination to do the job right. When the pink trenchcoats are on a job, it doesn't go belly up because the plan was bad, they don't have the skills, or they don't care about subtlety, it's because the magician just got splattered with knight errant blood and he hates seeing blood on his clothes and he just cannot even with the mission until he's cleaned it off, even if there's only 30 seconds until another patrol comes around the corner. It's because the samurai ate the decker's last can of SoySpam and now they're not talking but they've got to work together, so every communication sounds something like "Will you guys tell the samurai that she's about to step in front of a security camera and frag us all." "Will you tell the decker that disabling security cameras is his job and I shouldn't have to hide from them if he just turned them off."
That's my definition of pink trenchcoat.