What fallacy is this if any? Strawman, Definist, reframing?

If you don't believe something exists and you don't believe that something does not exist, I would suggest expressing that pair of non-beliefs as "I don't know whether that something exists". Anything else is prone to misconceptions, because when you say for example "I don't believe in vampires", it is in day-to-day conversations (something that does not apply in formal logic) implied that you believe there are no vampires. I think people, in general, say "I don't know" when they're uncertain rather than "I don't believe".   

I have had some success with saying, "You know how you know 'I don't believe you' isn't a claim?"

When I say, "God doesn't exist," it logically follows that you could respond with: "OK, prove it."

When I say, "I don't believe in God," it wouldn't make any sense to say, "Prove it."  You'd be asking me to prove that I don't believe in God, which is not something that could be done beyond your believing me to be honest or not when I make the statement.

[To Doug Glaass]

I don't agree with this, for a couple reasons. The definition of claim is pretty broad, simply that you are asserting something. That thing being true, or there being some viable way to prove it, isn't actually a requirement.

While a person's internal thoughts and feelings are difficult to prove, it can often be inferred from their actions, and you can build a base of evidence to support such a claim. For instance, if you tell me "I don't want to steal your dog" then I go to great lengths to ensure that you're never alone with my dog, it's pretty reasonable to infer that I don't believe you.

How are you going to prove whether or not I believe something.

"I don't believe [insert story]" is not an assertion of anything. It means I'm not convinced by the evidence.

If you come to me and say, "I took a submarine down to the Titanic."

I most likely would not believe you.  That wouldn't tell you anything about whether or not I think going to the Titanic were possible, the existence of the Titanic or the ability of submarines to go that deep. It would simply mean I don't believe you.

This is a really easy concept to understand except when the story that people say they don't believe is some man-made yarn about goat herders and where they thought the sun went at night.

When you say "I don't believe," you are asserting that you do not believe. The claim is that you do not hold the belief. Whether the underlying story is true or false, that is secondary.

Whether or not there is any way to prove that you hold that belief is completely irrelevant. If a claim can't be proven or disproven, it is still a claim. I can claim that, on this date 20 years ago, I ate a ham sandwich for lunch. As no records have survived, there's no way to prove or disprove that, but it doesn't cease to be a claim.

There are some circumstances when belief as a claim in and of itself is actually important, criminal law being one of them. In a self defense case, a defendant could claim that they believed they were in imminent danger. More broadly is the concept of mens rea. A defendant could claim that they did not believe their actions were illegal. In both instances, the prosecution and defense will present evidence, in an attempt to prove whether the defendant's actions were consistent with those beliefs, and by extension prove whether that belief was actually held.

[To Mr. Wednesday]

I'm glad we've had this exchange because I now understand the problem, and you've asked me to understand why people become confused when the existence - or not- of a deity is invoked.  Usually, the two issues haven't realized that they have not agreed on a mutual understanding of what "believe" means.

In your ham sandwich example, were I to say I don't believe that you ate a ham sandwich, i think it would be reasonable to infer that I, in fact, was of the opinion that you didn't eat one. A claim.

When I say (and I think many others) "I don't believe in God," what is heard is the meaning above and thus inferring the claim of the non existence of a Hod. However, what we really mean is, "I'm not swayed by the evidence I've this far evaluated. Nothing has convinced me yet that a God exists."

Going back to your sandwich, were I to say, "I'm not convinced that you ate a ham sandwich," in this case, I don't think it would be reasonable to think that I was of the opinion that you didn't.  You may have, but based on what I know, I'm can not with any certainty affirm your claim.

Much could be picked apart about this, but let me zero in on one issue that ruins the flow of conversation and stares out at me like broccoli caught in my dinner guest’s teeth. At the onset, you state that “I do not believe there is no God. I just don't believe there is one.” The fourth time that your interlocutor speaks he attempts to repeat your position back to you, but the words you have typed into his mouth reveal him as having heard you say “I do not believe there is no God, I just don't believe THEIR (insert x religion) is one.” He heard you as saying “their”, according to what you have typed into his mouth, whereas what you actually meant was “there.” Amidst the rest of the chattering, this discrepancy, appearing more than once, is not rectified at any point. 

I don't think the two statements are necessarily contradictory. Does a chair believe in god? No. That is not because the chair believes there is no god, but because a chair is an inanimate object and it doesn't believe anything. If you were to talk about a person rather than a chair, it would be unusual phrasing, but not technically incorrect.

Also to split hairs on the difference between atheism and agnosticism: The generally accepted consensus is that an atheist is a person who does not hold a belief that a god or gods exist. Agnostic is Greek for "without knowledge," an agnostic is a person who specifically holds the position that it is unknown whether god exists. That said, a person who believes there is no god, but that they can never know for sure, would be both an atheist and an agnostic. A person who has never heard of gods, and has never had the opportunity to turn ta belief either way, would be an atheist but would not be an agnostic.

I'm going to have to agree with the ai response. And that false dictomy as being one of the main fallacies. I forgot about that one. Indeed, just because you don't have a belief that something is true does not you believe that something to be false.

[To Jack]

Based on a more careful reading of this exchange, I'm going to take back my first paragraph. That said, I don't think this qualifies as a false dichotomy. In the second half of the exchange, starting with "Correct, but that isn't the case here," they are openly acknowledging that the third option, not holding a belief either way, is a possibility. The point they seem to be making is that the way you phrased your statement doesn't effectively communicate that you're taking that third option.

Like I said before, this is really just semantics revolving around the meaning of "I do not believe." By a literal interpretation of the words, you are correct that it could mean that you lack a belief. However, a native English speaker would typically use that phrase to mean that you believe the opposite (either firmly, or with uncertainty) and they're arguing that's the only possible interpretation.

Semantics bit is a good point. And mind you the person I was speaking English was not his native language. I'm also pretty certain that the statement I made was one of a formal logical statement. 

Even though it's about belief which is bit more nuanced and complex than mere binary values it still seems to read like a formal not statement/argument.

However the whole argument seems somewhat redundant. Wouldn't it be easier to just say " I do not have a belief that x exists?".