  We unify two existing approaches to the tau invariants in instanton and
monopole Floer theories, by identifying $\tau_{\mathrm{G}}$, defined by the
second author via the minus flavors $\underline{\operatorname{KHI}}^-$ and
$\underline{\operatorname{KHM}}^-$ of the knot homologies, with
$\tau_{\mathrm{G}}^{\sharp}$, defined by Baldwin and Sivek via cobordism maps
of the $3$-manifold homologies induced by knot surgeries. We exhibit several
consequences, including a relationship with Heegaard Floer theory, and use our
result to compute $\underline{\operatorname{KHI}}^-$ and
$\underline{\operatorname{KHM}}^-$ for twist knots.
