I'm with you...to long time Whovians, this is simply the following on from Hartnell, Troughton, etc (excluding the non-canon Cushing films, however fun they are). I would also - if we're being completist about things, include the radio series that McGann did between his 1990s movie and the Davies "reboot" (at least, that's what they call it).
I can't remember how many of those there were, but it was a cool way to keep the Doctor going when TV budgets couldn't allow for them to be visual episodes at the time. They were good audio productions, though, and had many crossovers to the TV show: it was clearly, like Trek's animated "fourth season", a continuation.
When Eccleston came along, the intention was a clear reboot, since we never saw a regen between McGann and the new Doctor, and I don't think he "died" at the end of his radio stint. There was a clear line drawn but the show then quickly reverted to referencing previous stories and encounters, and it was vaguely clear as it went on that this was the tenth, eleventh, twelth Doctor, etc.
He used to only have ten "lives", but they've upped it somewhere along the line to 13, which would be Smith's transformation into the next one. Lots of "make him a girl" or "make him black" rumors going around here, so it'll be interesting to see how they go, but what they really need to do is sort out the final regen problem. Of course, there's always a way in sci-fi (my personal "favorite" is that he regens into a former self somehow, however briefly, thus resetting a few lives, or that there's some kind of regen mishap that resets the counter).
Over here I've not watched the latest season, because I missed the second half of the last and I want to catch up with that first. But from what I hear the Doctor has accidentally infiltrated his own timeline, so this might be the get-out clause for a reset/solution to the recasting regen issue.
Nevertheless, as a time lord, the Doctor can't die: there will always be a version of him out there, somewhere in space and time. If he's clever, he'll divert some event so that he doesn't regen at that time and skews his timeline in a different direction. Yes, it breaks his own rule, but as I said, the show breaks these kinds of rules all the time.
The Doctor is an old, crotchety, alien being. Whoever plays him should reflect this.
