Ah, I think I see it now. Weak admissibility only requires a nontrivial periodic domain to have positive and negative coefficients if $c_1(\mathfrak{s})$ vanishes on it for the reasons you've explained, and makes no demands of periodic domains seen by $c_1(\mathfrak{s})$. Strong admissibility makes the same requirement of periodic domains on which $c_1(\mathfrak{s})$ vanishes, but now also requires periodic classes on which $c_1(\mathfrak{s})$ is positive to have some sufficiently positive coefficient (and doesn't care when $c_1(\mathfrak{s})$ is negative on the domain). Correct?

Thanks, the edit really clears things up -- and my question was admittedly unclear. I've accepted the answer since it fully handles the torsion case I asked about, but I'm also curious about the non-torsion case (where it's pointed out that strong admissibility still implies weak admissibility). My follow-up question won't fit here, so I've edited it into the original post. Would you mind taking a look?