I am not clear there is such an “important dissection [distinction?].”
If we are interested in the truth of any matter then regardless of whether a claim is made or not with reference to X, the truth or falsity of X is what ought to concern us.
It makes no sense, IMHO, to make a distinction between “I reject that the FSM exists,” and “I reject claims that the FSM exists.” The latter just seems academic if the FSM demonstrably does or does not exist. In fact, “claims" need not even concern us if the truth of the matter is what we are concerned about – talk of “claims” just adds a unnecessary layer.
Furthermore, “I reject claims that the FSM exists,” is actually meaningless without the rejection being grounded in justifiable certainty that the FSM, does indeed, NOT exist.
Onus probandi isn’t a philosophical concept, it is a means of off-loading responsibility for arriving at the truth – which is what ought to concern us, primarily – and a means of conveniently placing the burden concerning what we are not interested in onto others.
Never been much of a fan, since, in effect, it means using “I am not interested” as the philosophical equivalent of “Go suck a lemon!” It assumes that one’s own position is correct and the alternatives are not – at least, until someone else proves otherwise. But why is THAT up to someone else?
A good philosopher will anticipate alternative perspectives and work to either disprove or accept them, not counter with “The burden is yours!”
No, the burden is on anyone interested in the truth to get to bottom of any issue by assiduously considering all possibilities.