Is the induced electric field in a current changing solenoid actually that what we call magnetic vector potential field?

Question

Induced electric field $E$ in a changing current (i.e. changing magnetic field $B$) solenoid

image creditis: https://faculty.uml.edu//Andriy_Danylov/Teaching/documents/L18Ch33InducedEcovered.pdf

I was wondering if the induced electric field $E$ in that case is actually referring and identical to the magnetic vector potential $A$ field around a magnetic field $B$ source. Therefore, the mathematical concept in physics and electromagnetism of magnetic vector potential field $4$ can be physically interpreted, as what the induced electric field would be by a magnetic field if this was changing with time?

WP seems to support this interpretation in its given definition for the magnetic vector potential, quote:

In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: ${\textstyle \nabla \times \mathbf {A} =\mathbf {B} }$. Together with the electric potential $φ$, the magnetic vector potential can be used to specify the electric field E as well.

Answer 1

The magnetic vector potential isn't the induced electric field. However, in the case that the net charge density $\rho = 0$

The induced electric field can be expressed purely in terms of the magnetic vector potential.

$\vec{E} = - \frac{\partial \vec{A}}{\partial t}$

Infact, given the net charge density is not zero, this equation can also be used to find the induced solenoidal component of the E field, provided we are in the coulomb gauge.