now add reimagining and reboot
With the twin provisos that basically no one outside of ultra-nerd enthusiast forums cares about any of this type of distinction and that in reality many games don't fall neatly into exactly one category, I think it's relatively easy to make four useful categories that are distinct from each other:
Remaster: On analogy to a film or album remastering, a game remaster is taking something that already exists and reworking it to have increased fidelity. Increasing resolution, re-rendering FMVs, swapping models and textures for higher-res versions of the same design -- this stuff goes into a remaster, which should look essentially like a nicer version of the same thing, and play exactly the same as the older version. Things like FFX/X-2 for PS3/4, Perfect Dark for XBLA, and Another World 20th Anniversary fall in this category.
Remake: Taking a game and assembling it from scratch (literally making it again) without absolute dedication to recreating the precise experience, but with the goal of at least maintaining the core of what makes it the same game. The game might have a different style of graphics, or tweaked mechanics, or new systems or characters, but would generally retain enough to feel obviously connected to the original. REMake, Rondo of Blood and Tactics Ogre on PSP, Oddworld: New 'n' Tasty, Metroid: Zero Mission, and FF3 and FF4 on DS would all fall into this category.
Reimagining: Taking the concept of a game and making a new game based on it; maybe the levels are completely different, maybe it's the same story in a different type of game altogether, but the result is something that stands alone as separate and distinct from its inspiration. Tomb Raider Anniversary or the new Ratchet and Clank for PS4 fall here; they tell the same story(ish) as their inspirations, but don't try to maintain level design or gameplay systems.
Reboot: A game that tries to start a series fresh, abandoning the built-up storyline and mechanics and trying to begin again with a different and hopefully more appealing approach. Prince of Persia 2008; XCOM 2012; Tomb Raider, Shadow Warrior, SimCity, and Killer Instinct 2013 -- if you have to distinguish a game by pegging a pretty recent year in after its formal title, that seems to be a pretty decent indicator that it's a reboot.
After going through the majority of the posts I think the thread should be titled " ITT: Everyone has their own definition of what the word remake and a remaster is"
At very least, no company has used "Remaster" to mean anything with more upgrades than, say, FFX, so if you wanted to draw a line there and say everything below it is a remaster, everything above a remake, you'd probably have a 99% hit rate on what stuff actually gets labeled as.