Back again, Sterling! That last paragraph of yours gave me quite a headache!
The concepts you make use of in your question – belief, assertion, fact, proof – are all of them highly contestable concepts in the theory of knowledge. Philosophers have spent much time and ink trying to tease them out and there is no universally accepted final analysis. Despite this, I’ll make a suggestion and you can let me know what you think of it.
The core of your question seems to rest on a distinction between a belief and an assertion with you suggesting (if I understand you) that the conditions that attach to the one may not attach to the other. Let’s take the following as examples:
A: I believe there are fairies at the bottom of my garden, and
B: There are fairies at the bottom of my garden
Proposition A, which is a claim about my epistemological states, is true if indeed I do hold such a belief. That belief would still be true even if there were in fact no fairies at the bottom of my garden.
Proposition B makes no necessary reference to epistemological or psychological states. It is true only if there are in fact fairies at the bottom of my garden. However, if would be distinctly odd to add to B “…but I don’t believe it!”
A belief can be, as it were, ‘raw’ (“I believe X”; “Why do you so believe?”; “I just do”) or it can be ‘justified’ (“I believe X”; “Why do you so believe?” “Because I have seen the evidence for X”).
The epistemic distinction between a justified belief and an assertion (leaving aside their psychological or linguistic differences) seems to me to be vanishingly small. If that be so, then the conditions attaching to truth or proof for the one attaches to the other, other things being equal. I repeat: I may be wrong on this, and there has been much vigorous discussion on these matters in philosophical circles.
The general point in my discussion of the fallacy in question is that, other things being equal, one is normally entitled to believe or assert X only where there is evidence for X. Someone who denies X is not normally required to prove that there isn’t an X. It is sometimes said that it is impossible to prove a negative but this isn’t always so. If you claim that there is an elephant in the room I can normally prove that there isn’t by conducting an extensive and exhaustive search of the room.
While the burden of proof normally falls on the assertor/believer, it can be the case, as Wittgenstein says in his “On Certainty” that doubt needs grounds. Children learn the power of the question early on in life and use it to elicit response after response until eventually their interlocutors give up in exasperation!
Reading through my response, I’m not entirely sure I have addressed your concerns. If you’d like to continue the conversation, please do so.
Once again, apologies for the delay in responding to you.