What precisely do we mean when we say that an atom has a certain electron configuration?

For a hydrogen atom things are simple since it has only one electron. You can calculate the wavefunctions that are eigenstates of energy. We then say that these are the different shells and different orbitals (like s-orbitals, p-orbitals, etc.). In the case of an atom with multiple electrons we then say that the electrons fill multiple orbitals in a certain order which approximately follows the so called "aufbau principle". But what precisely do we mean by this electron configuration filling these orbitals? The orbitals are one-electron wavefunctions. You can construct a multiple-electron wavefunction from one-electron wavefunctions using a Slater determinant. However, that can't be precisely what we mean by an electron configuration because none of the eigenstates of energy (including the groundstate) of an atom with multiple electrons can't be described by a single Slater determinant. That would only be possible if the electrons didn't interact. You can approximate the groundstate with a single Slater determinant by e.g. using the Hartree-Fock method. Is the Hartree-Fock solution what we mean by the atom's electron configuration or something else?