Ohm's law tells us that current density $\vec{J}$ is proportional to electric field $\vec{E}$. But from my last post, as well as this blog, a comment introduced me to the concept that it is instead the surface charge density that created the electric field and helps in current conduction.

So which electric field is the $\vec{E}$ in $\vec{J}=\sigma \vec{E}$ here? Is it of the surface charge densities of the wire, or of the battery? Or is the electric fields inside the wire due to the surface charges the same as that of the battery? Kindly enlighten me with the proper concept.