Well said, Dustin; I've never felt that Dalton was a continuation of Moore.
Without getting into the whole 'same wife, same guy' debate, or the CR67 theory of a new agent adopting an old agent's name along with his number, I'd just like to say that I see Sean/George/Roger as one Bond in one timeline, and that the events in MR never really happened.
TLD was constructed in such a way that you can see it as a reboot if you want, or not. Right from Dalton's first appearance he exuded the attitude of a man with a dirty job, often called upon to do distasteful things that erode his soul - doesn't seem like a suddenly youngified Connery or Moore to me. "He was married once, but that was a long time ago" could mean anything. If you want to believe in an elastic timeline, then you can believe that - despite what it said on the tombstone on FYEO - that Tracy (or at least Dalton's Tracy) died more recently.
But just because we have a 'new' 007 it doesn't mean that another agent had his name changed from Cledwyn Jones to James Bond to 'confuse the enemy.' The series was refreshed in 1987, and this would have been even more apparent if a 34 year old Brosnan had taken over from Grandpa Rog.
Some people see the backstory PTS in GE as Brosnan 'erasing' Dalton from the timeline. Not me. I think Dalton and Brosnan were the same character. We see why Bond was so dour through the late 80s, and we see that he's lightened up a bit in the early 90s, at least until his past comes back to haunt him.
CR was obviously a reboot, but despite what Mendes said about coming 'full circle' it doesn't have to mean that Bond's next mission will be to investigate the goings-on at a bird sanctuary in Jamaica. The way I see it, Craig will continue to play the only version of Bond who's been provided with an actual origin story. The films may be reminiscent of the first two series - in the way that LTK was reminiscent of the novel LALD at times - and for the same reason: having mined the novels to near exhaustion, Eon is now free to mine the old films, without Bond visiting Istanbul 'again' or finding another woman smothered in a precious resource.
Lighten up, people - it doesn't have to all fit together. Craig is not going to be introduced someday to a familiar, elderly man as his 'successor.' Just as Roger Moore was not supposed to be the same Saint of the 1940s or his actual successor, neither is Bond supposed to be immortal or a succession of agents using the same name (although the new series coming this fall features a conversation between Sir Roger and Ian Ogilvy where they discuss how "the world needs a Saint", so perhaps they at least are related).
Lewis Wilson, Adam West, Michael Keaton and Christian Bale played different incarnations of Batman, but this doesn't mean that Batman (and Commissioner Gordon, et al) has been a constant in Gotham City since the 1940s. Just like Bond, they started over, without a standing history.
When Craig is done, I suspect that he will be replaced with another actor playing the same Bond (albeit with a somewhat different look and personality), as it will be too soon for another reboot. Provided the series (hopefully) continues for another few decades, another reboot is yet to come.
And on the debate will rage...