As an atheist I have been reading with interest some of the exchanges in this section of the forum. It seems a lot of theists are genuinely at a loss as to why some atheists insist that atheism is defined as "not believing in Gods" as opposed to "believing there are no Gods". I'm one of those who don't really care about the definition as long as people make it clear what their talking about. But in the spirit of peaceful co-existence(!) I thought I'd explain why I think many atheists are pedantic about this.
To my mind the distinction is not between negative and positive claims but probable and certain claims. An atheist who says "I don't believe in God" is always, in my experience, happy to say God probably doesn't exist (Dawkins even gives a number on a seven point scale). (The only exception I can think of might be the theological non-cognitivist who will say that they lack a belief in God because they don't know what the word means but won't make any positive statement about God because... they don't know what the means!) However if atheism is defined as Belief that God's do not exist the question of whether this assertion is made with 100% confidence or just on the basis of probability is left to the imagination.
Interestingly I saw one comment (from a theist) quite rightly expressing incredulity that anybody would imagine that atheism denotes the 100% certainty that God doesn't exist. (Sorry I can't find it right now). This theist quite rightly pointed out that no theist claims 100% certainty so why would anybody expect an atheist to express 100% certainty. I suspect that a lot of theists assume that atheism is the belief that God probably doesn't exist whereas ironically it is the atheists who think it might mean believe with 100% certainty. But why is this?
I think the reason is that a lot of apologetic and especially counter-apologetic does seem to assume that the atheist must demonstrate the lack of God with absolute certainty. Take the problem of suffering for example. I've seen WLC debate this many times and what I consider to be his "summary argument" which I have heard every time he speaks on this goes something like "if the atheist wants to prove that God would not have created a world with this much suffering they would need to show how a universe could be created with less suffering in it. If they cannot do this then they have failed to prove their contention that we would expect to see less suffering in our world if a good God exists". Now if the motion of the debate was "suffering proves the non-existence of God" this would be OK. But the argument comes out independent of the wording of the motion, eg "Why does God allow suffering" or "Does the existence of gratuitous suffering support atheism". Please note, by placing this particular challenge WLC is asking for what would be a cast-iron 100% demonstration that the God of the Bible cannot possibly exist. If I could show you blueprints for a better universe (which obviously I can't) there would be no question that the universe was not created by the Christian God. What is even stranger is that WLC in his introductory comments will always go out of his way to "acknowledge that the "problem of evil" is a problem, which does demand an answer". So his position seems to be that the problem of evil does initially seem to argue against the probability of an omni-benevolent God but that the argument can be dismissed because it fails to achieve 100% certainty.
This is not the only example of counter-apologetics working on the implicit assumption that atheism must be proved 100%. I can say that the idea of God knowing the future and having free-will seems to cause all sort of logical problems. WLC can say that if Molinism is true then this could provide a way out. So divine free will and omniscience could co-exist. Once having "achieved" this it is declared that another atheistic argument has "failed". Just as the problem of evil was "defeated". But if atheism is simply the belief that God probably doesn't exist then the suggestion that theodicies, or Molinism, or existence outside of time could be possible are not a reason to reject atheism. If all of these things were very, very highly "probable" it would be a reason to reject atheism. The assertion that all of them are possible is only an effective refutation of "certain atheism", not "probable atheism".
So to summarise I feel that theists who define atheism as "God probably does not exist" are doing so without any desire to misinform or be disingenuous. But that definition doesn't seem to fit with the challenges and hurdles put before the atheist in debates with apologists. Atheists pick up on this and the best way to avoid this confusion is to re-word "probably doesn't exist" to "I don't believe in". I acknowledge that this might allow a game regarding burden of proof but of course most atheists are very happy to provide arguments that can support the positive assertion form of the definition. It's just easier to say "I don't have to prove atheism because I'm not making the assertion" than it is to say "I don't have to prove atheism because I'm claiming probability, not 100% certainty".
Hope this helps