Coming in at the Christmas Special was a tough way to introduce a new Doctor. To be honest I felt the way RTD introduced Doctors wasn't as special as Mofs. RTD was kinda 'on with the show' and I have always felt those earlier seasons were more through the companion's eyes, not the Doctors. When 11 showed up there was a slight shift in the series and it was more about the Doctor and how he needed and saw his companions.
While I think you're correct about how it started, I felt like the change occurred before that point. To me, it started with the end of Martha's tenure. The Doctor on the Titanic and the Doctor meeting up with Donna for the second time were definitely through the Doctor's perspective. This trend continued through the loss of Donna, the specials, and the End of Time.