[quote=@Garattee]
EDIT: I realize that I'm deviating from dictionary definitions, these are just words loosely used to define conflict on message forums.
[/quote]

Well, my favorite thing to do is to quote the dictionary and then draw parallels to a situation. The problem is that you have a lot of people misusing words, and with enough misuse the meaning of the word actually changes. I'll use the weabo community as an example for this. Westerners tend to call hard core cartoon porn Hentai, and soft core stuff Ecchi. But Ecchi is actually what you call the letter "H" in Japanese, and is slang for Hentai. So while they mean the exact same thing in japan, they have different meanings elsewhere. This is the same thing with forum slang, it tends to have different meanings as you go from place to place. 

Disagreement, argument, and debate pretty much have the same meaning, but they have different flavors. Two people arguing facts "intelligently" would rather you viewed their exchange as a debate. Someone who is simply stating an opposing view point who doesn't wish to argue would likely tell you they are disagreeing. Argument has a mostly negative context, but they can all mean the same thing.

A flame war is simply the point where an argument (read: any of the above examples) has become toxic and the participants are being toxic, which is essentially what you stated. Though the exact point at which an argument becomes a flame war varies from forum to forum.