Kind of? The problem is that the show doesn't really do either very well. Going back to what I said before this trailer is really really good at picking out all the bits and putting them in order that makes it seem like DW is this amazing superheroic figure travelling through space and time, saving lives. But in the moment it's all drowned out in the noise of all the bullshit, and different writers having different visions of what the Doctor should be, and having to stick inside the case of the week format without a lot of attention to the arc of the doctor and his companions over the course of the season(s), leaving this sort of muddy mess.
Also, the different Doctor's all definitely have their own vibe around them so it's hard to pin down the one thing "the Doctor" should be. Eccelston was the damaged war vet trying to put the pieces back together, Tennant was the childlike wonder seeing everything there is to see and sharing it with his friends that he absentmindedly puts in harms way along the way, and Smith is the fairy tale manic pixie dream boy that swoops in and saves you and shows you how amazing the universe is. I just have a personal affinity for the "mythical legend of the Doctor" version.
EDIT: I think the better word for what I'm thinking of would be the Gandalf Doctor. He shows up one day, can do all sorts of fantastic and amazing things, takes you on an adventure that changes your life forever or helps you save the world, and then he vanishes again leaving you and the world slightly better than it was before.