Degree-one cohomology of metacyclic $2$-groups: a low-bit dichotomy

We study the subalgebra generated by degree-one mod-$2$ cohomology for metacyclic $2$-groups $G=\langle a,b\mid a^{2^α}=1,\ b^{2^m}=a^c,\ b^{-1}ab=a^r\rangle$ with unique normal forms, where $α,m\ge2$ and $r\equiv-1\pmod4$. Presentation compatibility forces $c=0$ or $2^{α-1}$ and $G^{\mathrm{ab}}\co