"Refuses to accept that there's a God" seems like a definition that presupposes theism. We usually don't accept something that exists just because we don't like it. For instance, liberals don't accept Donald Trump even though they know he's all too real. (To accept something is to receive it, so it must be given in the first place.) That's what the presuppositional theists say: everyone knows God exists, and theists like God whereas atheists don't. That's a very biased way of understanding the distinction.

It's not just that atheists don't accept God. They deny that there's any good reason for thinking that God exists, regardless of anyone's feelings about the matter.