What is the difference between an $electrostatic$ and a $non-electrostatic$ electric field?
An electrostatic field is an electric field that is produced from a non-changing charge distribution with no current (i.e. $\mathbf J=0$ at all points in space for all time). Essentially for electrostatics we just have stationary charges and constant electric fields.
If this is not the case, then resulting electric fields are not considered to be a part of electrostatics. i.e. moving charges, EM waves, etc.