It doesn't. It simple renders the whole concept irrelevant to an atheist. Degrees of strength apply to belief. Lack of something is simply lack of something, not strong or weak lack of something.
I admit there are a handful of atheists who appear, frankly, evangelical about it. I am amused a bit by this, and perhaps irritated. I think they have taken bait that they should not have.
Tangentially...I think that, given the definitions of "strong atheism" et al., the strongest of the strong atheists are also strong theists. There are none so sure that Thor does not exist, as those who are certain Yahweh does. And, probably, vice versa. I mean, given that each theistic belief is separate, shouldn't each lack therof be separate? (no, not serious, pointing out the ridiculous position) So we have a-Thorists, a-Yahwehists, a-Zeusists, etc., do we not? And Yahwehists are a-Thorist, a-Zeusist, and a-alltherestist, far more "strongly" (in the sense of the definition) than any plain vanilla atheist ever was (possible exception of Dawkins).
And what if you were absolutely certain of the non-existence of many, but not all, gods? You perhaps give a 20% shot to each of the Hellenic Pantheon, but are not certain--but are 100% certain that Thoth is a fictional character? What do we label you?
Sorry, the whole "strong, weak, yadda yadda" label is couched in terms that really are irrelevant to me. I will continue to pass.