I think this gets back to the question of what you mean by "there". Because if I have, say, water in a tank, and I move around a stick I placed in that water, then the 'water field' (or whatever I want to call the positions of the water molecules) will update based on that, and it will update at a finite speed because the information is carried by traveling water molecules. So the water field is there because water molecules are there- if you put something in their way, they'll run into it.
But electromagnetic waves are carried by photons, which are really weird. Water molecules have a rest mass- if you managed to slow one down to no speed at all, it would exert about as much gravitational pull as normal, and it would still get in the way of other things you tried to push through it. A photon has no rest mass, and a way of thinking about that is to say that if the photon isn't moving, it isn't there.
And so if by "thereness" you mean "if I shoot a neutron at a stationary one, is the neutron sometimes deflected?" then water molecules are there and photons aren't.
But there's another sense that we can talk about thereness- what happens when we (or they) speed up. If I had the aforementioned water tank on a train moving near the speed of light, things would look the same inside the train- but really weird from outside. To observers, the ability of the 'water field' to update depends on how fast the water field is moving relative to the observer- but that isn't true for the electromagnetic field. All observers see it 'updating' at the same rate.
So, what do we mean by "aether"? Here, I think we might be getting in a linguistic/historical issue, which is what you originally asked about. I was fascinated, watching a talk between PZ Myers and Dawkins (one of those, might not be the first one), where for Dawkins the phrase "group selection" seemed to be irretrievably connected with Wynne-Edwards, despite there being several defensible things that also have a connection with that name. Each time it came up, he had to check- "you're not talking about Wynne-Edwards group selection, right?"
I believe (but haven't extensively researched the physics in question) that the luminiferous aether was tied to the idea that there is one correct reference frame, and so when that disagreed with experiments that idea got tossed out. That means we don't have a visual answer to the question "what wiggles when there's an electromagnetic wave?", and as far as I can tell it doesn't make a difference what you visualize wiggling, but calling it 'aether' makes people ask questions to make sure you're not an adherent of a dead theory.
Related to: Dissolving the Question, Words as Hidden Inferences.
In what sense is the world “real”? What are we asking, when we ask that question?
I don’t know. But G. Polya recommends that when facing a difficult problem, one look for similar but easier problems that one can solve as warm-ups. I would like to do one of those warm-ups today; I would like to ask what disguised empirical question scientists were asking were asking in 1860, when they debated (fiercely!) whether atoms were real.[1]
Let’s start by looking at the data that swayed these, and similar, scientists.
Atomic theory: By 1860, it was clear that atomic theory was a useful pedagogical device. Atomic theory helped chemists describe several regularities:
Despite this usefulness, there was considerable debate as to whether atoms were “real” or were merely a useful pedagogical device. Some argued that substances might simply prefer to combine in certain ratios and that such empirical regularities were all there was to atomic theory; it was needless to additionally suppose that matter came in small unbreakable units.
Today we have an integrated picture of physics and chemistry, in which atoms have a particular known size, are made of known sets of subatomic particles, and generally fit into a total picture in which the amount of data far exceeds the number of postulated details atoms include. And today, nobody suggests that atoms are not "real", and are "merely useful predictive devices".
Copernican astronomy: By the mid sixteen century, it was clear to the astronomers at the University of Wittenburg that Copernicus’s model was useful. It was easier to use, and more theoretically elegant, than Ptolemaic epicycles. However, they did not take Copernicus’s theory to be “true”, and most of them ignored the claim that the Earth orbits the Sun.
Later, after Galileo and Kepler, Copernicus’s claims about the real constituents of the solar system were taken more seriously. This new debate invoked a wider set of issues, besides the motions of the planets across the sky. Scholars now argued about Copernicus’s compatibility with the Bible; about whether our daily experiences on Earth would be different if the Earth were in motion (a la Galileo); and about whether Copernicus’s view was more compatible with a set of physically real causes for planetary motion (a la Kepler). It was this wider set of considerations that eventually convinced scholars to believe in a heliocentric universe. [2]
Relativistic time-dilation: For Lorentz, “local time” was a mere predictive convenience -- a device for simplifying calculations. Einstein later argued that this local time was “real”; he did this by proposing a coherent, symmetrical total picture that included local time.
Luminiferous aether: Luminiferous ("light-bearing") aether provides an example of the reverse transition. In the 1800s, many scientists, e.g. Augustin-Jean Fresnel, thought aether was probably a real part of the physical world. They thought this because they had strong evidence that light was a wave, including as the interference of light in two-slit experiments, and all known waves were waves in something.[2.5]
But the predictions of aether theory proved non-robust. Aether not only correctly predicted that light would act as waves, but also incorrectly predicted that the Earth's motion with respect to aether should affect the perceived speed of light. That is: luminiferous aether yielded accurate predictions only in narrow contexts, and it turned out not to be "real".
Generalizing from these examples
All theories come with “reading conventions” that tell us what kinds of predictions can and cannot be made from the theory. For example, our reading conventions for maps tell us that a given map of North America can be used to predict distances between New York and Toronto, but that it should not be used to predict that Canada is uniformly pink.[3]
If the “reading conventions” for a particular theory allow for only narrow predictive use, we call that theory a “useful predictive device” but are hesitant about concluding that its contents are “real”. Such was the state of Ptolemaic epicycles (which was used to predict the planets' locations within the sky, but not to predict, say, their brightness, or their nearness to Earth); of Copernican astronomy before Galileo (which could be used to predict planetary motions, but didn't explain why humans standing on Earth did not feel as though they were spinning), of early atomic theory, and so on. When we learn to integrate a given theory-component into a robust predictive total, we conclude the theory-component is "real".
It seems that one disguised empirical question scientists are asking, when they ask “Is X real, or just a handy predictive device?” is the question: “will I still get accurate predictions, when I use X in a less circumscribed or compartmentalized manner?” (E.g., “will I get accurate predictions, when I use atoms to predict quantized charge on tiny oil drops, instead of using atoms only to predict the ratios in which macroscopic quantities combine?".[4][5]
[1] Of course, I’m not sure that it’s a warm-up; since I am still confused about the larger problem, I don't know which paths will help. But that’s how it is with warm-ups; you find all the related-looking easier problems you can find, and hope for the best.
[2] I’m stealing this from Robert Westman’s book “The Melanchthon Circle, Rheticus, and the Wittenberg Interpretation of the Copernican Theory”. But you can check the facts more easily in the Stanford Encyclopedia of Philosophy.
[2.5] Manfred asks that I note that Lorentz's local time made sense to Lorentz partly because he believed an aether that could be used to define absolute time. I unfortunately haven't read or don't recall the primary texts well enough to add good interpretation here (although I read many of the primary texts in a history of science course once), but Wikipedia has some good info on the subject.
[3] This is a standard example, taken from Philip Kitcher.
[4] This conclusion is not original, but I can't remember who I stole it from. It may have been Steve Rayhawk.
[5] Thus, to extend this conjecturally toward our original question: when someone asks "Is the physical world 'real'?" they may, in part, be asking whether their predictive models of the physical world will give accurate predictions in a very robust manner, or whether they are merely local approximations. The latter would hold if e.g. the person: is a brain in a vat; is dreaming; or is being simulated and can potentially be affected by entities outside the simulation.