I also thought that Pence would be the President behind the scenes. I think that this is what Pence thought too.
I’m not so sure now. Trump has two conflicting impulses. He is ignorant, totally incurious and unwilling or unable to do the work that the office of the President demands.
The other side is that Trump is a complete egomaniac and narcissist. He will not share the limelight with anyone.
The Vice President only has responsibility for the things that the President allows. Trump will cut Pence off if it starts to get out that Pence is acting like the President. Trump wants to take all of the credit and all of the attention, with none of the blame. So Pence will have to walk a fine line of exercising influence while avoiding notice.
Dick Cheney’s health problems meant that he could not run for President, even if he wanted to (and it’s not clear that this is what he ever wanted). Cheney also had a lot of government experience. This allowed Cheney to work behind the scenes.
In contrast, I think that Pense wants to run for President. He may even realize that his association with Trump makes this problematic, but it was his best option. This may mean that he is not as able or willing as Cheney to work in the shadows.
So far it looks like what Trump may do is set up warring camps that all have nebulously defined responsibilities. Pence may be one of these camps. But there will be at least one other camp that has power that vies with Pence.