electromagnetism - How to read Maxwell's Equations?

I was talking with a colleague about Maxwell's Equation, in particular Faraday's Law of Induction, and I realised that I did not understand the following. Usually, this equation is expressed as:

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit. - Wikipedia

In other words, a magnetic field that varies in time generates a circular electric field. Now I tried to express the same idea but backwards, that is, a circular electric field generates a magnetic field that varies in time proportional to the curl of the electric field. My colleague did not agree with this idea and said that the only interpretation is the one equivalent with the statement from Wikipedia.

This got me thinking and the reasons to back up my point of view are:

1. Since the relation in discussion is $\nabla \times \vec E=-\partial_t \vec B$ one should be able to read it both ways, since the $"="$ sign follows the properties of an equivalence relation


2. If I consider the relations $\vec D=\epsilon_0 \vec E$ and $\vec B=\mu_0 \vec H$, we have the induced fields $\vec D$ and $\vec B$ as functions of the field $\vec E$ and $\vec H$. The way I understand the term "to induce", it would much rather make sense to say that Field A generates The induced Field B, than The induced field B generates Field A. Going back to Faraday law, this would mean that the electric field generated the induced magnetic field.

Although the second point is based more on my understanding of the terminology, which might be subjective, I do not find the first one to have the same issue.

Which is the correct way of looking at these equations (same problem arises on Ampere's law too) or is there a book/written material that addresses this topic, that I can find?

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Edit: So far the question received two answers that are based on the causality of the problem, both of them suggesting that the electric and magnetic fields should be considered as a connected thing (thus the electromagnetic field). I did not search for material suggested by AlbertB, and will do in the following hours, but I have a follow up for both answers.

If there is no delay when it comes to "generating" one field from another, because they are intertwined, wouldn't that mean that an electromagnetic wave should propagate with an infinite velocity? This question neglects relativity, and I consider having a flawed image on how an EM wave propagates.

Answer

It appears that there are two phenomena intermixed, and that might be the source of the confusion. Both, Rob and Albert, are talking about the properties of electromagnetic waves. While you are talking about the properties of current and charges, independent of each other.

If you have a wire loop(s) being "crossed" by a magnetic field, the magnetic field will induce a current in the wire, which creates a voltage difference at its end points.
If you have a voltage source at the end point of a wire loop, the electric field will induce a magnetic field perpendicular to the loop.

From these examples, one can see that the processes (therefore the equations) are reversible.