Non-Raider: I think this is the most common usage for casual WoW players. While most people admit there are casual raiders, using just the word "casual" often connotes someone who does not raid. This version of casual player may not have time for raiding or may not want to raid or may be too busy playing alts to ever get to max level.
Non-Hardcore: This encompasses both casual raiders and those who don't raid. As Amanda Miller said while we were discussing this in our virtual offices, it is more about defining what "hardcore" is. This extremely NSFW animation of a famous Onyxia wipe pretty much defines what most people consider hardcore: a lot of yelling and penalties when you don't follow your guild's strict rules. Hardcore players choose their professions, talents and gear according to the dictates of their class officers and raid leaders. Hardcore guilds pursue progression aggressively, hopefully leading to realm firsts. Anyone who is not this serious is often considered casual. (Ready Check has a great discussion of hardcore and the conflict between hardcore and casual.)
Noob: Many "serious" players consider casual to be synonymous with noob. There are certainly players who are new to the game (or just the content they are currently exploring) that are casual. But there are also new players who consider themselves pretty hardcore. Of course, "noob" doesn't refer only to new players, it is a more general term referring to anyone who is less knowledgeable than the speaker. We all began as noobs, regardless of whether we are casual or not, so this is just a bad definition of casual and one only used pejoratively.
Players with Lives: This is the other side of the coin. Some casual players feel superior to hardcore players because they say that hardcore players don't have lives. There are many serious raiders who are really good at prioritizing their lives, hold down steady jobs, etc. There are also many casual players who are failing at balancing real life and WoW, even though they never raid. Your success at real life is not governed by your play-style, nor is your play-style defined by your success at real life.
The real way to determine who is casual is by the individual's definition of fun. A raider's definition of fun is to get together with others, down bosses, better their gear and accomplish world/realm firsts when possible. Because of this definition, someone with better gear and more raiding accomplishments is a better player than someone else who agrees with the definition, but hasn't accomplished as much.
A casual's definition of fun is just as rewarding to the individual, even though it is usually easier to achieve. We want to get the most fun out of the time we have to play -- whether that is by roleplaying, PvP, playing multiple alts, creating our own achievements, leveling with friends and family, etc. The casual player may find it fun to read all the quests, stop and take pictures of the scenery and do each quest one at a time. Many of us just want to leisurely level our favorite characters while enjoying the camaraderie of friends, whether by questing together or just chatting in guildchat. You cannot tell which casual player is better than another since their definitions of fun are different.
And that is the best definition of casual that I can come up with:
Casual players cannot be rated better or worse than other players.
Hardcore players and raiders have ways to compare their level of play to each other. They can be rated as "the best" or "scrub" according to their Armory page and in-game skill reputation. Ratings just do not apply to casual gameplay. No casual player is better or worse than any other casual player. "Oh yeah! Well, I enjoyed my questing in Borean Tundra more than you did!" Comparisons are just irrelevant.
But that doesn't make a casual player better than a non-casual player. We are all having fun -- or should be. Otherwise, why are we playing? So you hardcore types, keep enjoying your gear/skill/achievement/progression comparisons. You should be proud of what you have accomplished and that you are having fun doing it. And we casual types should be proud as well. We are having a great time doing what we like to do and relieving the stress of our daily lives at the same time.
And don't feel bad when someone tries to rain on your parade by calling you names and making comparisons. Funsuckers exist in real life, too, but unlike WoW, you can't just put them on /ignore and never hear from them again. It's too bad really. Real life would be a lot more relaxing if you could.