I have very vague memories of being a 6 year old watching a very energetic man with white hair and a yellow "Bugsy" automobile that can fly. That was my first encounter with The Doctor and to be honest I was confused by the show.I was probably too young to understand what was going on.
Eccelston was my first true introduction to The Doctor as a mature adult. What I saw was a man running about as if he's the savior of the world while carrying a terrble burden. It didn't make sense, I did not get why anyone would follow this man around nor do I feel his presence. At least for me, Eccelston's Doctor felt more like an ordinary human.
With Tennant something just clicked into place somewhere in the middle of his first season and I could identifiably say, "Now that's The Doctor". This is a being that is to be respected, to be listened to, to have fun with. This is why I consider David Tennant as "my doctor". Having said that, when I think of the scene that exemplifies Tennant's doctor, it always comes back to him saying "I'm sorry. I'm so sorry". Its not a good image.
Smith on the other hand, just screamed "The Doctor" to me from the first brief moment I saw him in "The End of Time". The months of waiting for his first season as The Doctor to begin was agonizing. He has since lived up to that expectation very well despite me disliking the vignette style story telling that's being used in major episodes.