While the whole atheist-agnostic thing is more of a continuum than not, I think the line is whether you believe there is no God or whether you don’t believe there is one. It might seem pedantic but I do think it’s an important distinction.
I used to identify as agnostic for a while--the definition I quoted is one adapted from The Atheist Experience podcast. I'd personally describe someone who actively believes there is no god, i.e. possesses enough evidence to claim "There cannot be any god" an "anti-theist", in my personal lexicon.
Generally this is why I bother to give the specific definition by claim--too often I've encountered religious folk who take "atheist" to mean "one who hates god", or "one who actively has evidence against a god", or similar--which may be true for other folk who use the label atheist, but doesn't match my personal definition.
As you said, "the whole atheist-agnostic thing is more of a continuum than not," so I like to give the definition to clear up any mis-assumptions people may make about me for using said label :)
I also agree that the difference does matter--I also think it's more intellectually honest of me to allow for the possibility that I am wrong, however unlikely--even then, there is an important distinction between acknowledging the existence of a god, and choosing to worship said god if they even existed (which is usually the follow-on question from religious folk I talk to.
not OP, but I would presume "the claim" is that (a) there is a God, (b) the church knows which one, and (c) they know what he wants.
For comparison, I (atheist) normally state my position as "There is insufficient evidence to conclude there is a God," so any statement about there being any god, or about what influence they should/do have on our lives I treat as a "claim," which requires supporting evidence.