Inhomogeneity in the ground state is an intriguing, emergent phenomenon in magnetism. Recently, it has been observed in the magnetostructural channel of the geometrically frustrated $α$-NaMnO$_2$, for the first time in the absence of active charge degrees of freedom. Here we report an in-depth numerical and local-probe experimental study of the isostructural sister compound CuMnO$_2$ that emphasizes and provides an explanation for the crucial differences between the two systems. The experimentally verified, much more homogeneous, ground state of the stoichiometric CuMnO$_2$ is attributed to the reduced magnetoelastic competition between the counteracting magnetic-exchange and elastic-energy contributions. The comparison of the two systems additionally highlights the role of disorder and allows an understanding of the puzzling phenomenon of phase separation in uniform antiferromagnets. ; Comment: Published version, freely available at http://www.nature.com/srep/2015/150319/srep09272/full/srep09272.html