I googled for "definition of film remake" and pulled this up:
"In cinema, a remake is a work that has the same story, and often the same title, as a work that was made earlier. The term is generally used in reference to a movie which uses an earlier one as the main source material, rather than in reference to a second, later movie based on the same source. For example, 2001's Ocean's Eleven is a remake of the 1960 film, while 1989's Batman is a re-interpretation of the comic book source material which also inspired the 1966 movie."
Seems to me that our beloved Casino Royale fits right in with the Batman example. CR 2006 is a re-interpretation of the Fleming source material which also inspired the 1967 movie.
We have our answer:
A re-interpretation of the Fleming source material.
Google don't lie.
Well I don't know about Google, but I would agree with that quote above completely. That's precisely what I thought of the Batman movie, both 1989 and 1966. They were both very different, but could arguably be said to be similar enough by those in favour of calling it a remake. Both had the Joker as a villain and both involved some nefarious plot which Batman had to foil.
Of course, the word "re-interpretation" gives the connotation that this version is putting their own spin on the source material, different to the way it was interpreted in the 1967 movie. However, while that may be technically correct, I would say that the 1967 movie more qualifies as a re-interpretation (in that it re-interpreted the source material instead of following it straight), and this 2006 movie as a "proper or faithful interpretation".
But that's all just semantics. I think the Google quote above nicely summarises the gist of the idea.