I've seen this same debate multiple times on here. I don't know why people would care if Bob should be considered an atheist or not? I don't really. It is just a trivial semantic issue but perhaps I'm missing some important issue someone else sees.
It was important enough for you to weigh in on it.
For me, it leads into another issue: the assumption of monotheism as a default position.
Nobody could reject all god-concepts ever conceived by humanity, so when people talk about defining atheism in terms of rejection, the required process usually means rejecting some sort of narrowly-defined "standard" god-concept (typically, their own religion's god), and then make an exception for people who believe in "non-standard" god-concepts.
Under the "rejection" model, it's never the rejection of some fringe god that makes a person an atheist: when someone has rejected Thor but doesn't have an opinion about Yahweh, the "rejection" crowd never says that this rejection qualifies as atheism. OTOH, they're happy to call someone an atheist if they've rejected Yahweh and have no opinion about Thor. It's based on a double standard.
It also mischaracterizes the positions of people that are generally accepted as atheists... or at least conflates rejection of arguments for the rejection of the conclusions being argued. Plenty of atheists who have put significant thought into the issue hold a position something like "all of the arguments I've ever heard for gods are junk. If any gods do exist, it isn't for the reasons I've been given, and I ought not to accept a god-claim without good reason to do so." This isn't really rejection.
I also think a big part of the "rejection" definition is to paint atheists as unreasonable by implying that they jump to conclusions without evidence.