Why is it said that light can travel through empty space? Here are a few commonly heard sentences that will make my question clear:
Statements #1
“The Michelson Morley experiment once and for all did away with the ether or the idea that light needs a medium in which to travel”
“Light can travel through empty space”
Statements #2
“Light travels though the electromagnetic field”
“The photon field is only one of the fields for each particle in the standard model.”
“There must be dark matter in the universe for our current model of gravity to make sense.”
“Space is not empty. A quantum foam fills it”
You can see the confusion that arises from the conflicting statements which are commonly heard in physics.
The fact that light travels via an electromagnetic field is in clear contradiction to the statement that light travels through empty space. I suppose we could try to join empty space and the EM field into one thing by saying light is the field. Frankly, that sounds like doublespeak.
What am I missing?
Light is a disturbance in the EM field. There is no empty space. Light travels through a field.
Why do some physicists say light can travel through empty space?
 A: When it is said that light can travel through empty space, it is meant that it does not need matter to propagate.
A: I would agree with and repeat the idea that an electromagnetic field is a mathematical construct, not a literal reality, but that doesn't mean it is void of meaning and that you should just "shut the eff up and calculate", either. It is a term in mathematical language that lets us translate and integrate a wide variety of statements such as "this room has a lot of light in it", "there is red light shining here", and more and many more we cannot give straightforwardly in English at all, all into one neat expressive construct. But it definitely refers to real things - you just have to be careful as to what real things, and not to say the "word" itself "exists" out there as a reification. Its relevance to the question is that you can translate the question into this mathematical language and express it there. The reason that doesn't answer it is the same reason asking it in, say, Spanish, doesn't answer it.
But I also say the question is not unanswerable, either. Because there is an answer, and that answer lies in empirical evidence: we say light propagates through empty space because we so far have not found that taking anything out of space that would otherwise be there injures the light or impedes its transmission in any way. Instead, it is the presence of things in space that harms transmission. Though we, of course, cannot at least with out existing technology remove all subtler constituents like dark matter from an area of space, but we have no evidence that to do so would impede light - indeed, were light's propagation to depend on the amount of dark matter present, then it would not be "dark" at all!
A: I guess what a lot of people mean when they say that light travels through empty space is that it doesn't require a physical medium (matter) to travel through - like sound waves do.
The fact of the matter is that light does travel in the electromagnetic field, but the electromagnetic field is just a mathematical tool - as are all the other fields in QFT. You can't touch the EM field, just like you can't touch the fields that give rise to physical matter. You might say that we can touch matter and so those fields are more real. However, the sensation of touching those fields is just a result of electromagnetic interactions between the particles in ourselves and those in the object we are touching, and the reason for that being true is no more real than the reason that light can move from A to B.
To answer your specific points, the Michelson-Morley experiment proved that light travels at a constant speed for all observers. It proved that there was no physically-measurable medium in which light travels through (like sound). The electromagnetic field is how we describe light moving, however, it is not a physical object - it is just a mathematical tool and so only exists on paper.
I think the confusion in terminology comes from people trying to attach physical meaning to fields. Maybe you can attach meaning to them, and one day we will be able to look at them in a different way to how we can now. However, it must be noted that the fields introduced in QFT are mathematical tools, and that is all they are. We should not imagine them as filling space like a fluid, they simply make useful and accurate predictions.
In summary, they tell us how everything works - they will never tell us why.
A: I think it's likely due to the fact that for many centuries, "vacuum" and "empty space" had the same meaning.  The vacuum was seen as something that had absolutely nothing in it.
Now, "vacuum" is defined as space that contains no matter (but might contain non-matter).
A: The EM field permeates all of space (even in vacuum), just like all other fields like the gravitational field and the others.
The photon is the excitation of the EM field, and this excitations propagates as a wave when we talk about EM radiation. Emphasis is on the radiation.
So even when we are in vacuum (even like in the voids of intergalactic space), the EM field still permeates all of it. It is not empty. It is said as you say that light can travel through this void because the fields all permeate it.
What is confusing as you say is that people usually talk about media that are required for certain waves like sound to travel through like sound travels through air or water. Air and water are made up of particles of matter, and in everyday life we call these media.

The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves.

https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment
You mention the Michelson-Morley experiment, and it proved that there was no ordinary medium (meaning made up of matter particles) that light would travel through.
The other very nice example of this is gravitational waves, they too do not need any physical medium (made up of matter particles) to propagate, since they themselves are the stretching and squeezing of spacetime itself. This radiation travels through empty space of course, because it is the wave that exist and propagates in spacetime itself.
So yours is a beautiful example because today we know that not only EM waves but gravitational waves too propagate through vacuum without the need of a medium (made up of matter particles), just because both are propagating excitations and distortions of the underlying fields and spacetime itself.
A: 
Why do some physicists say light can travel through empty space?

What a physicist means by this is that there exist non-trivial vacuum solutions to Maxwell's equations. A vacuum solution is one where the sources are zero. In the case of Maxwell's equations that means that the charge density and the current density are zero. Additionally, the specific solutions are also solutions to the wave equation and waves are often described as "traveling".
Of course, "there exist non-trivial vacuum solutions to Maxwell's equations which are also solutions to the wave equation" is not a description that will really help most people. So instead we say "light can travel through empty space". This indeed conveys the idea well, with an unavoidable loss of technical precision. It is a fine non-technical statement, to be understood without too much nit-picking.

saying light is the field. Frankly, that sounds like doublespeak.

I think that is actually the correct statement. The idea that light travels through the electromagnetic field is strange. If I say that a plane travels through the air that means that there is some distinction between the plane and the air. They are different things and one is physically located in the other. That does not correctly express the relationship between light and the electromagnetic field.
Light is a specific type of solution to Maxwell's equations, which defines electromagnetic fields. Specifically light is a vacuum solution with travelling waves (perhaps even specifically waves with wavelengths in the optical part of the spectrum). As such, light is indeed an electromagnetic field and there is no in-principle distinction between it and the field. So saying that light travels through the electromagnetic field is more like saying that wind travels through air than that a plane travels through air. You can say it, and you are not completely wrong, but it is strange.
