Why isn't the electromagnetic field considered as the medium for electromagnetic waves by scientists in general? According to a google search, the definition of a medium in physics is the substance that carries a wave. The definition of a substance in physics is matter with specific composition and properties. The definition of matter in classical physics is a substance (this is confusing. Is substance a type of matter, is matter a type of substance, or are the two terms interchangeable?) that has mass and takes up space.
So an electromagnetic field meets the definitions of both substance and matter. As a substance, it has a specific composition as it is made of an electromagnetic field (the most fundamental entities are made out of themselves) and it has well defined, specific properties. As matter, it produces a gravitational field and hence has gravitational mass, and also has inertial mass, as seen here:
https://en.wikipedia.org/wiki/Electromagnetic_mass
It also occupies space.
Given all this, why isn't the electromagnetic field accepted as a medium for electromagnetic waves by the scientific community?
Note: the aether theory had a substance pervading all of space as an absolute frame of reference and medium through which electromagnetic waves traveled, and movement relative to it (such as earth's orbit around the sun) would cause the speed of light to change.
The special theory of relativity removed the need for an absolute frame of reference, as light (and all massless entities in general) always move at c, regardless of movement relative to them.
But I do not understand why this removed the need for the electromagnetic wave to have a medium. Special relativity removed the need for aether as an absolute frame of reference, but it did not disprove of it, and by pure logic, a wave must have a medium to propagate at all, right? So can we call the electromagnetic field an 'aether' for electromagnetic waves?
 A: First - don't get too hung up on dictionary definitions of extremely broad concepts.  I've been a student of physics for quite some time now, and I don't think I could give an exact definition of "matter."  I know what the dictionary says, but there a lot of things which we'd call matter that don't fit that description.  In fact, the dictionary, while wonderful for normal English, is a terrible resource for physics students, so I'd altogether avoid trying to reason about physics by waving definitions around.
Now, on to your question.  You ask why the electromagnetic field is not accepted as a medium.  The answer is that when we say that electromagnetic waves do not require a medium to propagate, what we mean is the following.
Prior to the discovery of electromagnetic waves, the waves with which we were familiar took the form of disturbances in some material, such as air, water, or rock. When we speak of the speed of such a wave, we are implicitly working in a reference frame in which the bulk material is at rest.  If the medium is moving with respect to us, then we will observe a higher or lower wave speed depending on the direction of its motion.
For electromagnetic waves, this is not so. There is no such material, and there is no such reference frame.
Now, you may ask why you can't just say that the electromagnetic field is an intangible medium with no sense of rest frames, but at this point you're just playing with terminology.  We specifically say that electromagnetic waves don't require a medium precisely to mean that there is no "background stuff" which carries the wave and with respect to which we are moving (or not).

Special relativity removed the need for aether as an absolute frame of reference, but it did not disprove of it, and by pure logic, a wave must have a medium to propagate at all, right?

An absolute frame of rest is incompatible with what we would now call the special theory of relativity.  Also, "pure logic" is not really a good tool to use, and in this case, the answer is apparently no.
A: 
So an electromagnetic field meets the definitions of both substance and matter.

To clame the existence of a physical entity it always will be good practice to measure this entity. What one can measure is an electric field from separated subatomic particles, say an excess and a lack of electrons on the two plates of a capacitor. Furthermore a magnetic field one can measure for the aligned orientation of the magnetic dipoles of subatomic particles (see how works a permanent magnet). In both cases a test specimen of the same nature will behave as the test body. Electric fields could be measured with electrical charges, magnetic fields with magnetic dipoles.
But how about an electromagnetic field? For me an EM radiation could be set up only by the emission of photons. Ask yourself which other possibilities do you have to produce EM radiation without disturbing electrons (and other particles). Of course a stream of photons could be described over its intensity and direction and a swelling characteristic for modulated radiation. But is such a radiation a force field like an electric or a magnetic field? What are the test specimen? Photons? Of course you could measure the temperature increase of an illuminated body and the pressure on a body from the momentum of the photons. But could you draw field lines to illustrate the behavior of an EM field?

But I do not understand why this removed the need for the electromagnetic wave to have a medium.

Firstly photons are moving as a selfinductance of an electric and a magnetic field without any medium. They carry energy that the get from disturbed particles and which the give to other particles when they hit them. In between they are indivisible units on their own.
Secondly there is a spatial dependence of their behavior from the gravitational potential. Seen from a distant observer the local velocity of light slows down near Black holes and is higher far way from big masses. So a gravitational field influences the movement of light. I could claim this an aether but I want not get in trouble and so I wouldn’t.
A: Water does not move, waves move.
EM field does not move, the wave moves.
Field is the lowest possible state of energy.
Particles are its excitation and is localized, hence particle.
When you further excite the particle, that excitation move as waves through the already existing field.  Empty space is not empty.There is permitivity and permeability as constant value.
These constant values is what dictates the speed of propagation.
You can stick to the definition of background as some material (whatever that means) as the medium only to eventually have to face that the so called material themselves are made of that underlying field which one tossed as a medium which placed one standing on shaky ground.
In the meantime you still have to deal with particles being point like , that is it doesn't occupy spatial dimension which puts you back on the hot seat again.
When it comes to it in physics field is assumed to exist throughout empty space but not waves or forces.  Fields don't move, waves do.  You decide.
