You are in effect proposing that weak atheism (a well-established term) should not be considered atheism at all.
I consider the "weak" and "strong" distinctions not particularly clear.
I prefer "agnostic atheism" and "gnostic atheism" instead: it's the difference between "I believe that gods don't exist" and "I know that gods don't exist", with the former being "weak atheism" and the latter being "strong atheism".
Leaving aside that I disagree, I wonder if you think a substitute term should be used, or if you would rather not acknowledge the stance at all. Or maybe there is a third choice?
I wouldn't consider most people who have the opinion that gods don't exist to merely have a "lack of belief". I think they are the same as me: they disbelieve that gods exist. It's an opinion and a position, not a lack of one.
In your thread I made a list of those that I'd consider merely having a "lack of belief" in gods: babies, people completely ignorant of the topic, and people who haven't cared enough to consider the topic (and therefore, can't have come to a decision on what they believe or don't believe).
I think the stance that you are talking about has been incorrectly defined. Basically, the definition came first-- because of its ease of defense in debates and maybe from some squeamishness about acknowledging that you believe that gods don't exist-- and then people were convinced that their beliefs fit this easy-to-debate definition. In other words, it's a made up stance that doesn't accurately portray people's positions.
As mentioned above, I think "agnostic atheism" and "gnostic atheism" more accurately present the types of atheism. I wouldn't consider those who merely lack a belief, as defined in my list, as atheists. If they need a label, which I don't think they really do, then I think that "non-theist" would be the most appropriate (though non-atheist would be just as accurate).
EDIT:
Forgot a group in my list. "Agnostics" of the popular sense. While they have considered the topic, they don't know which is more likely; the arguments for both sides could be said to be equally convincing; they too have not come to a decision on what they believe yet.