I had to stop at the "burden of proof" bit.
"Since it is impossible to empirically verify that every single thing is not God, then one cannot prove that no thing is God. "
^is just a flawed statement. God is a specific thing, a shoe on your head is a specific thing. It's quite easy to disprove most definitions of god, just like you can disprove a shoe on your head (though not
that easily with god, usually). It only takes a small number of observations to disprove most definitions of god (e.g. an omnipotent god that gives a sign, absence of that sign), it sometimes takes more than that to prove them (e.g again, a god that gives a sign, as the sign could have come from a different source), and the rest tend to be unprovable definitions (e.g. "god is outside the visible universe entirely") or definitions without a proposition (e.g. "god is the universe").
It simply does not follow that checking a single object to see whether it is a god, affects the probability of a god existing. Whether the universe is finite or not, one does not have access to the probability of god existing based upon how many objects they have already observed, even if you knew how many objects were left to observe. Access to the probability usually comes from specific observations and knowledge of the nature of that god.
There is a difference between
atheist = somebody who believes there is no god
and
atheist = somebody who does not believe there is a god.
The two definitions are often used alongside, but the difference is vital to this discussion!
The point you and Jonny make regarding this is fine. I'd say though, that those two sentences you chose do not differentiate between the two types of Atheism put forward. So, if you said one of those sentences to someone, they would not be able to tell which type of Atheism you meant. I could tell because the point was already familiar to me, and I could read your sentences in a formal manner, but in general English those sentences say the same thing.
"Absence of belief" and "belief in absence" are more clear, I find.
An Agnostic does not say it is impossible to prove one way or the other. That is only a specific type of Agnostic.