I'm reading Griffith's text, and he starts by defining Electrostatics as requiring the source charges don't move. I've seen a few slightly different definitions of electrostatics and magnetostatics. If you wanted to start from the full Maxwell equations in a vacuum, how would you precisely define Electrostatics and Magnetostatics? Would Electrostatics be the condition that $\frac{\partial\vec{B}}{\partial t}=\vec{0}\implies\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_o},$ and $\nabla\times\vec{E}=\vec{0}?$
And would Magnetostatics be the condition that $\frac{\partial\vec{E}}{\partial t}=\vec{0}\implies \nabla\cdot\vec{B}=0,$ and $\nabla\times\vec{B}=\vec{0}?$
If so, how would you conclude from the electrostatic equations that the source charges don't move? I can see if you add in the requirement that $\frac{\partial\vec{E}}{\partial t}=\vec{0},$ but if you're only given $\frac{\partial\vec{B}}{\partial t}=\vec{0},$ how do you see this?
One of the other definitions for magnetostatics I've seen is $\frac{\partial\rho}{\partial t}=0.$ If magnetostatics is the condition that $\frac{\partial\vec{E}}{\partial t}=\vec{0},$ then can't you see $\frac{\partial\rho}{\partial t}=0.$ from $\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial t}(\nabla\cdot\vec{E})=\nabla\cdot\frac{\partial\vec{E}}{\partial t}=0?$