  We prove that the stability condition for Fano manifolds defined by
Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow
weight, is equivalent to the existence of anticanonically balanced metrics.
Combined with the result by Rubinstein-Tian-Zhang, we obtain the following
algebro-geometric corollary: the $\delta_m$-invariant of Fujita-Odaka satisfies
$\delta_m >1$ if and only if the Fano manifold is stable in the sense of
Saito-Takahashi, establishing a Hilbert-Mumford type criterion for $\delta_m
>1$. We also extend this result to the K\"ahler-Ricci $g$-solitons and the
coupled K\"ahler-Einstein metrics, and as a by-product we obtain a formula for
the asymptotic slope of the coupled Ding functional in terms of multiple test
configurations.
