  Using a numerically exact first-principles many-body approach, we revisit the
"prototypical" Kondo case of a cobalt impurity on copper. Even though this is
considered a well understood example of the Kondo effect, we reveal an
unexpectedly strong dependence of the screening properties on the
parametrization of the local Coulomb tensor. As a consequence, the Kondo
temperature can vary by orders of magnitude depending on the complexity of the
parametrization of the electron-electron interaction. Further, we challenge the
established picture of a spin-$1$ moment involving two cobalt $d$-orbitals
only, as orbital-mixing interaction terms boost the contribution of the
remainder of the $d$-shell.
