I think both what you get in real time and what you get when someone has time to consider and check for errors and polish are "real". It's not that each person has a perfectly well-defined set of beliefs and ideas, and (1) you get to see them unfiltered in real time and only see a fake version when the person has time to polish, or (2) you get to see them accurately when the person has time to make sure they're getting it right, and only see a rough draft full of mistakes in real time. People are more complicated than that.

Some people are good at getting things right in real time. Some people are good at getting things right when they're careful. I think it is a mistake to call someone stupid because they're bad at one of those things -- unless for some reason one of them matters more than the other (e.g., they're in a position where they have to make quick decisions all the time; or they are primarily a writer who can always think and check and polish before publishing).

I'm not convinced that the definition-of-atheism thing has much to do with the real-time versus polished distinction. But:

I think both atheists and theists sometimes cherry-pick definitions for tactical purposes. "Oh no, I can't possibly be expected to offer evidence that there are no gods, because being an atheist just means not positively believing in gods." "Of course so-and-so's evil actions don't reflect badly on Christianity -- if he were a real Christian he'd be acting better." "Why should I care what liberal Muslims say? The most authentic Muslims are obviously the ones who behead people while shouting Allahu akbar." Etc., etc., etc.

But it's not silly to define "atheist" as "not positively believing in any gods"; that's a property that many people have, it's perfectly reasonable to have a name for it, and the only problem with using "atheist" for it is that most people use that word to mean something a bit different. And the trouble is that there aren't "enough" words; we've got "atheist" and "agnostic" and "deist" and "theist" but there are more gradations of belief than that. (Absolutely certain there are no gods. Of the opinion that on balance there are probably no gods. Of the opinion that whether there are gods is unknowable in principle. Of the opinion that it's knowable in principle but one doesn't in fact have enough evidence. Inclined to think that there's probably something godlike but we can't know anything much about it. Thoroughly convinced that there is something godlike, but that we can't know much about it. Somewhat convinced by arguments for a particular religion but still unsure. Firmly committed to a particular religion and confident that it's right. Etc.)

Personally, I use "atheist" to mean "overall of the opinion that there are no gods, whether or not certain of this", "non-theist" to mean "not positively believing in any gods", "agnostic" to mean "substantially unsure whether there are gods, whether or not one thinks it's knowable in principle", "theist" to mean "overall of the opinion that there is at least one god, whether or not certain of this", and I qualify those terms in whatever ways might be necessary if I want to say that someone's certain there's no god or kinda-halfheartedly-Hindu or whatever. I am somewhat prepared to argue that all those choices are better than the alternatives. But if someone else only uses "atheist" and "theist" for people who feel completely certain about whether there are any gods, or uses "atheist" to mean "not positively believing in gods", or something, that's a defensible choice so long as they take the trouble to be clear about what they mean and refrain from cheating by equivocation.