A DC current through a inductor does not create electric fields
An electromagnetic field is created when a current passes through a wire

This is entirely correct.

As far as I know, since the current is DC, there is a magnetic field (since current is moving), but there is no electric field

It is true in a certain sense, and under certain conditions, that a DC current through a wire creates a magnetic component of a electromagnetic field, but no electric component(*) of an electromagnetic field.
Let's parse this. In the 1830's Faraday published the results of experiments he had performed with magnets and coils of wire. What he discovered is that the relative motion of a magnet and a coil of wire induced a current in the wire. I want to emphasize relative, because the story eventually involves Einstein and the Special Theory of Relativity.
In the 1870's Maxwell published a theory of electricity and magnetism that could either be described as theory of a single field, an electromagnetic field, having two components, an electric field component, and a magnetic field component. Or, could be described as two separate fields that influence each other in clearly defined ways.
There was a problem with Maxwell's account, however. Faraday had pointed out that it was relative motion between the magnet and the coil that induced the current. However, according to Maxwell's theory, a stationary magnet induces no electric field, yet a movement of the coil, causes current to flow. (Think of a generator with stationary permanent magnets, and a moving coil).
Lorentz saw the need to add an additional equation to the ones formulated by Maxwell, (which were recast as we know them today by Heaviside). Lorentz added a &quot;Lorentz force&quot; equation, that allowed a current to be induced in a coil moving through a stationary magnetic field.
While this accurately predicted what would happen, Einstein was not happy with this state of affairs. What had been a unitary phenomenon for Faraday, the relative motion of a magnet and coil inducing a current, had become with the electric and magnetic field theory, two separate phenomena. Either induction of an electric field by a changing magnetic field, or a Lorentz force, depending upon one's frame of reference.
Einstein changed this by asserting a single electromagnetic field, which could be divided into electrical and magnetic components in different ways depending upon one's frame of reference. What is &quot;magnetic&quot; in one frame of reference is &quot;electric&quot; in another, and vice versa. Within a frame of reference in which the coil is stationary and the magnet moving, the electric field component can be seen. Within a frame of reference in which the magnet is stationary and the coil is moving, the electric field component is absent. But, as they say, it's all relative.
Your question does not involve a moving magnet. But the principle is the same. In one frame of reference, there is no electric field component, but in another frame of reference there is.
So, it is entirely correct that:

An electromagnetic field is created when a current passes through a wire

And, in the frame of reference where the wire is stationary, but only in that frame:

since the current is DC, ... there is no electric field

or more properly

Since the current is DC, there is no electric field component(*) of the electromagnetic field.

*[minor point. All of the above goes on the assumption that the voltage drop through a wire is negligible.]