How are the "Classes" - A, B, or C, players defined? On this website, the following ratings are given:


  
  1200-1399 = 'D' player - usually a beginner
  1400-1599 = 'C' player - average club or tournament player, most people can achieve this level if they work at it
  1600-1799 = 'B' player - consistently above average
  1800-1999 = 'A' player - strong club player, takes the game far too seriously!, has lots of opening knowledge
  2000-2199 = 'Expert' - extremely strong, consistent player with the possibility of achieving Master rating, may have real talent.
  2200-2399 = 'Master' - strongest amateur rank, hasn't quite got the hang of things yet but maybe on day he/she will wake up.
  
  
  International professional players have two ranks:
  
  
  2400-2499 = 'International Master' - weakest professional rank; strong, experienced international player, eats Masters for breakfast
  2500+ = 'Grandmaster' - eats IMs for breakfast, lunch and dinner, a star in the firmament of Caissa, a chess genius who thinks nothing of playing 20 and 30 board simuls against Experts and Masters and is disappointed if he/she doesn't win every game, capable of playing 10-20 blindfold games at the same time, and winning, etc. etc, in short, an all around bricks and mortar, brass bound b*st*rd of a player, but they do lose on occasion, sometimes to players with a much lower rating and computers are better than that these days.
  


Note that these ratings are for "USCF". When I see these kinds of "levels" of player on this site, is this the common and correct definition of what they refer too? If not, what is the correct definition?
The definitions you post are more of less correct though some levity has been inserted. The descriptions for the under-2400 ratings are consistent with the USCF.

IM and GM titles are not awarded by the USCF. They are awarded by FIDE and are based not upon rating but on achieving certain results at recognized tournaments.  That is, if one scores well enough at a FIDE tournament (based upon the competition), one earns an IM or GM "norm".  When one earns three norms, one is awarded the title.