This is absolutely the last go-around I'm having on this.

1. My definition of a belief is that any proposition you accept as true counts as a belief. For such a belief to be knowledge, it has to meet a few additional conditions; it has to be true, one has to have good evidence, etc. So every proposition that is known is also believed, but not every proposition that is believed is also known. This distinction seems extremely straightforward, no more confusing or vague than the distinction between squares and rectangles.

As far as I know, "belief" is not a technical term of art in physics, chemistry or biology, so I'm not particularly sure why their use of the term should count so much. In sciences such as psychology and cognitive science, the generally accepted use of "belief" is the one I've articulated. For instance, this Psychology Today says, "Simply, a belief defines an idea or principle which we judge to be true." in a 2001 research paper (pdf link), psychologist Simon Baron-Cohen defines the 'Theory of Mind' as "...being able to infer the full range of mental states (beliefs, desires, intentions, imagination, emotions, etc.) that cause action." This is fundamentally similar to how belief is understood in the context of the continuing debates over folk psychology. And hell, even the dictionary seems to agree with me. The Oxford Dictionary defines belief as "An acceptance that a statement is true or that something exists."

There are good reasons for the broader definition of belief. For instance, one may want to ask, "What is necessary for a belief to be a case of knowledge?" If belief is by definition something that isn't known, the question is pointless, but if it's defined in the more neutral way I've outlined, then it's an interesting and fruitful one. Or, are moral beliefs intrinsically motivating? Obviously, both beliefs that you know and irrational beliefs can play a role in motivating and influencing behavior, so there's not much point to adopting a more restrictive definition of belief in this context. Examples could be multiplied all day, but honestly, it's getting late, there's way more shit I have to get to in this post, and I'm putting off way too much stuff to write all of this.

If you didn't understand the point of the "all of John's children are asleep" example, then obviously you didn't understand what I said. Let me take this from the top. If you've done any logic at all, you know that "Some S are P" does not entail "Some S are not P." Nonetheless in ordinary conversation, if I say "Some of John's children are asleep," then pretty much everyone will infer that some aren't. Why does this happen, if it doesn't logically follow? H.P Grice's explanation is that people are tacitly following unwritten rules of conversation, one of which is "Make the most informative statement that is relevant." If all of John's children are asleep, then "Some of John's children are asleep" would not be the most informative relevant assertion I could make; since people generally assume that everyone else is following these tacit rules, they infer that "Some of John's children are asleep" was the most informative assertion I could make, in which case, it must be the case that some of his children are awake. This connection happens frequently and regularly enough that people start to assume "Some S are P" does entail "Some S are not P," which causes them to fail their logic class.

My claim is that the same thing happens in the case of knowledge and belief. When I say "I believe that p," people assume that I'm making the most informative relevant assertion possible. If I had known p, that would probably have been relevant, so since I said "I believe that p" rather than "I know that p," everyone will assume that I don't know p. Because this relationship is so consistent, many people will end up thinking that "I know p" entails "I do not believe p," but this does not follow from the propositions themselves. It's an artifact of the conversational rules that we are tacitly following.

2. Eh, I still sense some goal post shifting. It seemed to me that the point of your original thought experiment was to show that there could be no genuine agnostics (i.e. people who are roughly 50/50 in terms of their confidence that God exists). I'm not sure what being able to choose one's beliefs has to do with this at all. At any rate, I could simply revise my previous counterexample: Suppose we're presented with three doors: A either gives you 1 million dollars or causes you to lose an arm; B either gives you 1 million or causes you to lose a leg; C is a way out. Wouldn't everyone be 'compelled' to open door C?

3. The problem with defining atheism as a mere "lack of belief" is that it's too vague; on this definition, someone who is completely on the fence counts as an atheist; after all, they don't have belief in God. It also leads to some potentially dangerous intellectual traps: Someone who identifies as an atheist in this way has the option to vacillate between the weaker and stronger interpretations of the claim; in one moment, such an atheist can confidently proclaim that the whole idea of God is bullshit, but in the next moment, when challenged, simply insist "Well I didn't say that God does not exist, I just said that I have no belief in God." This is intellectually dishonest, and we want our terms to make this sort of dishonesty more difficult.

Do I need to say that I simply "lack belief" in Russell's teapot, or lack belief in ghosts, conspiracy theories, etc.? Or am I entitled to be reasonably confident that none of these things are real? If someone challenges me on my disbelief in Russell's teapot, must I retreat to the claim "Well I'm not making any positive claims! I'm not saying it isn't real"? I say no, and the reason has to do with intrinsic probabilities. I don't have much time to defend the idea, but the basic thought is that propositions have intrinsic probabilities (that is, probabilities independent of other evidence). Quick example: "George is a bank teller" is more intrinsically probable than "George is a bank teller and a cannibal." There are many more ways the world could have turned out that would make the first proposition true than would make the second proposition true (because every such scenario that would make the second statement true would also make the first one true, but not vice versa). Similarly, I think there are more ways for "God does not exist" to turn out true than for "God does exist" to turn out true. "God exists" is less intrinsically probable than "God does not exist," so therefore theists, but not atheists (even on the strongest possible interpretation of the term) bear a burden of proof. Again, I don't have the time to develop the idea in more detail here, so I'll just this there for now.

Lastly, I think there is plenty of positive evidence to justify the conclusion that God does not exist, regardless of how the burden of proof swings. There's the pervasiveness of extreme suffering, the fact that billions of people do not believe in or have much awareness of the God of western monotheism when we would expect a perfectly loving being to make himself known to his creation, evidence of poor design in the universe, etc. I think atheists would be well advised to move past concerns about the burden of proof and push these lines of evidence for the conclusion that God simply does not exist, and that we can be pretty confident of this.