How would you define electrostatics and magnetostatics starting from Maxwell's equations?

I guess different authors use different definitions. For me, it is that the E- and B-fields do not have time derivatives, hence curl free, conservative E-fields and B-fields that can depend only on steady currents.

The condition that the divergence of $\partial {\bf E}/\partial t = 0$ is not the same thing. The E-field could be time variable and have this still be true - e.g. in a transverse electromagnetic wave! Clearly that is not a magnetostatic situation either.

The curl of the B-field does not have to be zero in magnetostatics; steady currents are allowed, which obviously means you have to have (uniformly) moving charges. As ${\bf J} = \rho {\bf v}$, then $\partial {\bf J}/\partial t = 0$ implies only that ${\bf v}\partial \rho /\partial t + \rho \partial{\bf v}/\partial t = 0$. So it might be possible to arrange static magnetic fields by having a non-zero rate of change of charge density balanced by accelerating charges to somehow keep the current density constant! The continuity equation, $\nabla \cdot {\bf J} + \partial \rho/\partial t =0$, tells you that a time-varying charge density would require a current density divergence.