'Electrostatic field of a circuit' What does it mean? I've recently gotten into electrodynamics, I've studied electrostatics & magnetostatics, so far I know these facts:
Electrostatic fields are produced by static charge configurations
Magnetic fields are produced by moving charge configurations (currents)
Now there is something that I cannot understand, for example, a wire carrying current that is supposed to be neutral, but according to this statement:

There is an electric field inside the wire, which I know there must be, to actually drive current through the wire, but the wire is neutral, meaning it is not charged, so where does that electric field come from? Does it come from the source (battery)? And is at an electrostatic one, or does it come from a somehow induced charge within the wire, it could be the battery and that would easily answer the question. But what about the electric field outside the wire? Is it also due to the battery?
 A: Yes, it is true that the current carrying while is neutral. This is because the net inflow of charge is exactly equal to the net outflow of charge (electrons in this case). But there is a very a subtle distinction between the origin of the electric field inside the wire and outside it. 
You are correct in stating that the electric field inside the wire is created by the potential difference supplied by the battery. The electrons, in response to this field, move through the wire, producing a steady current.
But the electric field outside the conducting wire is not due the battery. It is due the electrons moving through the wire. Here's a link for more info on this: What are the fields produced around a current carrying conductor? 
So to conclude - The electric field created (inside the wire) by the battery causes the electrons to move, and the moving electrons create another electric field (outside the wire) while moving up the potential difference. 
Hope this helps!
A: Why do you assume the wire is exactly neutral?
On surface of conductors there can be stable distribution of electric charge that is not zero everywhere. If there is constant electric field inside the wire at some point, the surface distribution of charges in neighbourhood of that point is such that it produces that field. The closer the charges are to that point, the greater their influence on electric field there.
The charge distribution on the surface of wires is maintained by the electric field of the battery.
A: 
There is an electric field inside the wire, which I know there must
  be, to actually drive current through the wire, but the wire is
  neutral, meaning it is not charged, so where does that electric field
  come from?

The electric field within the wire must be established by the addition of, and redistribution of electric charge on the surface of the conductor.
Note that it isn't quite right to assume that the wire is electrically neutral.  Indeed, in an electric circuit, it is generally not the case.
For example, consider the simple case of a battery connected to a resistor by two conductors (assume the resistance of the conductors can be neglected). The (electric) potential of the conductor connected to the positive terminal of the battery is higher than the potential of the conductor connected to the negative terminal.  Clearly, at least one of the conductors has non-zero charge density.
Indeed, for there to be energy flow from the battery to the resistor, there must be an electric field from the more positive conductor to the more negative conductor. Associated with the flow of energy is a non-zero Poynting vector which requires both an electric and magnetic field in the space between the conductors:

