What does electrostatic/magnetostatic approximation mean, exactly? If I understand correctly, the electrostatic approximation assumes that all charges are stationary (i.e the charge density is constant in time, and current density is zero). The magnetostatic approximation assumes the current density is constant in time. Are these the correct definitions?
I have also seen electrostatics being characterized by an electric field which is constant in time. Is this supposed to be some obvious consequence of the above definition?
Looking at Maxwell's equations, the divergence of $E$ is fixed, but the curl of $E$ might vary in time if $B$ varies in time. Even in empty space (an electrostatic situation by defualt) we have the wave solution to Maxwell's equations, in which both $E$ and $B$ vary in time. 
So I'm guessing most authors are just sloppy when it comes to nomenclature, and when they say electrostatics, they really mean quasi-electrostatics, i.e electrostatics in the quasistatic approximation where the deplacement current $j_D$ is neglected from Ampere's law. 
In quasi-electrostatics, we definitely can't have the wave solution in empty space anymore. But is it still possible that the $B$ field varies in time? The magnetic field equations reduce to $curl(B)=0$ and $div(B)=0$. How do we know it's not possible to find some time-varying $B$ field which always has zero curl and zero divergence? 
 A: This are Maxwell Equations:
$$
\begin{align}
\nabla\cdot\mathbf E = \frac{\rho}{\epsilon_0}, \quad\quad
&\nabla\cdot\mathbf B = 0\\
\nabla\times\mathbf E = -\frac{\partial\mathbf B}{\partial t}, \quad\quad
&\nabla\times\mathbf B = \mu_0\left(\mathbf J + \epsilon_0\frac{\partial\mathbf E}{\partial t}\right)\\
\end{align}
$$

*

*In the electrostatic case: $\nabla\times\mathbf E = 0$.


*In the magnetostatic case: $\nabla\times\mathbf B = \mu_0\mathbf J$.
So, In the static cases, we do not have the other field changing. That is, in the electrostatic case, magnetic fields cannot change. In the magnetostatic case, eletric fields cannot change.
An electric or magnetic field is static when it is uncoupled. Not to be confused with spatially-constant field, or non-time-varying field. On normal conditions (Maxwell Equations), the sources which generates the electric field are $\mathbf B$ and $\rho$. That is, the electric field depends on what the magnetic field is doing, that is, they are coupled.
The whole point of the static approximations is to neglect field coupling. In the electrostatic case, a magnetic field can never generate an electric field. Meaning, the only sources of electric fields are $\rho$. The same is valid for the magnetostatic case: Only $\mathbf J$ generates $\mathbf B$, that is, $\mathbf E$ cannot produce $\mathbf B$.
Notice that, comparing Maxwell's equation with the respective approximations, we get:

*

*Electrostatic case implies: $\frac{\partial\mathbf B}{\partial t} = 0$ (it does NOT imply $\frac{\partial\mathbf E}{\partial t} = 0$ as one might imagine).


*Magnetostatic case implies: $\frac{\partial\mathbf E}{\partial t} = 0$ (it does NOT imply $\frac{\partial\mathbf B}{\partial t} = 0$ as one might imagine).
A: It appears that you've put the cart in front of the horse: the fact that the field is static means that the charges are stopped or moving slowly and that charge and current densities are mostly constant.
The time varying Maxwell Equations (which couple the $\mathbf E$ and $\mathbf B$ fields) are :
$$
\begin{align}
\nabla \times \mathbf E &= -\frac{\partial\mathbf B}{\partial t} \\ 
\nabla\times\mathbf B &= \mu_0\left(\mathbf J + \epsilon_0\frac{\partial\mathbf E}{\partial t}\right)\\
\end{align}
$$
In physics, we set time derivatives to zero for "static" cases. This may indicate that those derivatives are actually zero, or it may indicate that they are simply too small to consider. 
$$\frac{\partial \mathbf B }{\partial t} = \frac{\partial \mathbf E }{\partial t} = 0$$
meaning electrostatics studies $ \{ \nabla \times \mathbf E = 0, \nabla \cdot \mathbf E = \frac{\rho}{\epsilon_0} \}$ and magneto-statics studies $\{\nabla \cdot \mathbf B = 0, \nabla \times \mathbf B = \mu_0 \mathbf J\} $
There are no waves in electro or magneto-statics. If a field has zero curl and zero divergence, it should be obvious that it is constant (see Kelvin-Stokes Theorem).
A: I would say the definition is that the fields should be constant, not the charges. After all, an electromagnetic wave in a vacuum has constant charges and currents, and it's surely not electrostatic.
