The primary qualifier to be considered a remake instead of a reboot is following the same loose plot. It's the difference between Thunderball and Never Say Never Again vs Neverending Story > Sharkboy and Lavagirl. One is the same story being told in a different way. The other is a different story that transparently lifts elements from an earlier story. That thin line between homage and massive ripoff. (Aka, 90% of the storyline in Mass Effect and Halo.)
Tomb Raider: Anniversary is a remake of Tomb Raider that makes some pretty significant design changes. It's still the same loose story and setting, but with heaps of alterations.
Far Cry: Instincts was a remake of Crytek's Far Cry for Xbox. It has major, major plot and gameplay differences. But it's still a remake. It's someone else making the same game in a different way.
An example of this that comes to mind is how EA's Bond games sometimes had two (or more) games from different developers. Gearbox's Nightfire and Eurocom's Nightfire are completely different games. They're basically... "adaptations" of a shared source material created by EA in the form of concept art and all that stuff. They look and play very differently. But if they were released years apart they would be considered remakes of each other.
Let's imagine the film Quantum of Solace were a videogame, right? Then two companies "remade" that videogame. Treyarch's "remake" was literally Call of Duty. It used the CoD engine and played like CoD. Eurocom's "remake" was a third person shooter with heavy stealth elements. Very different game. Both games follow the same loose plot, but they play very differently.
I know some people are purists, but sometimes games are made better by genre shifts. It really comes down to the game.
