Bush was Governor of Texas BEFORE becoming President of the United States, and Cheney had nothing to do with that. And people who covered the Governor's beat are saying that the management style in the Oval Office is pretty much what it was in Texas.
Bush is described as a boss who expects his underlings to do what he tells him. He delegates a lot of work and his people do a lot of processing of information for him -- but he always makes the final call, and expects his people to toe the line. Those who don't are fired.
Apparently he doesn't have a problem with staff his staff disagreeing with his decisions, but he does expect every single one of them to toe the line publically and to do what he says.
I'm not privy to what goes on behind the closed doors -- and I doubt anyone else here is -- but I do believe that Bush Jr is calling all the shots. I think Cheney is probably a trusted advisor and that his opinions hold a lot of weight, but I do not believe he's the "real president," any more than I believed that Bush Sr was an idiot when he was president (that was sort of a popular opinion back then, too).
And that is why I'm more concerned than ever about things. Bush Jr is a guy who, having made up his mind, will simply not listen to anyone to disagrees with him. It scares me, a lot.