The distinction between “belief in” and “belief that” is certainly valid. I was taught the same thing in a philosophy of religion class, and I lean on that distinction in an upcoming article.

To say that anyone who makes a claim has some burden of proof is likely to say that we should be responsible for everything we say or do. That’s a fine general principle. (Or maybe your teachers were theists who were trying to poison the well against atheism.) The question is whether negative claims are claims in the relevant sense, or whether everyone’s burden of proof is equal. If there are different kinds of claims, there may be different burdens of proof.

Then again, legal principles might influence epistemology since the prosecutor and the defendant don’t have the same burden of proof in a court of law. One has to make the case, while the other only has to knock it down or cast reasonable doubt on it. In my article I argue that the legal tradition shouldn’t dictate the answer in epistemology.

And what I try to show is that epistemic burdens are, in practice, relative to background, default assumptions, or to what we take for granted. This is why folks today would expect much more from you (or they’d think you’re crazy) if you said that Zeus exists, than they would if you said the same thing in ancient Greece when Zeus’s reality was taken for granted. The burden of proof clearly changes as a sociological matter, as the background assumptions shift. Some things we take as obvious while others are more tentative, and the burden of proof changes accordingly.

Just because that’s how things do work, doesn’t mean they ought to work that way. Maybe in an ideal world, every proposition would be treated equally, but that wouldn’t be a human world.