Even if the distance through vacuum is different, but the distance through a medium (or simply just reflected) is the same?
Yes! Even then!
It would only look simultaneous if the speed of light in a vacuum is in fact instantaneous wouldn't it? Which is what we're trying to determine.
The medium is not relevant. Or, more accurately stated, it just adds a needless variable.
The speed of light is definitely different in different mediums, but you still have to somehow know when the beam left the source and when it arrived at the target. You can't know that information at a speed faster than the speed of light - regardless of what mediums the light beams went through or the route they took through them.
In other words, to detect a delay between two paths (ABD vs ACD), your camera or sensor would need to be synchronized to know when each beam left A, not just when they arrived. That’s the same as trying to measure the one-way speed of light.
Even though your setup uses a single sensor at D, both beams still pass through different legs of the journey, and your interpretation hinges on knowing when the light was at points B and C, which you can’t determine without a synchronized clock at A. That requires a simultaneous frame of reference which cannot be established without assuming the one-way speed of light in advance.
It’s truly an impossible thing to measure, because the act of measuring it requires you to assume it. You're forced to beg the question, since the speed of the signal delivering the information is the very thing you're trying to measure. It’s like trying to measure a ruler with itself. It’s a circular operation that reveals nothing.
You would only need two mirrors in the setup, positioned equidistant from the camera, but at different distances from the laser.
The idea is to isolate the second-half of the trip that light takes (vis a vis "two-way" speed of light).
You have two paths.
A is the laser.
B and C are mirrors along the path of the laser, with one cutting the laser beam in half.
D is the camera/sensor.
A -> B -> D
A -> C -> D
Both B -> D and C -> D are the same distance, but A -> B is not the same distance as A -> C.
If the one-way speed of light is c, and not instantaneous, then there should be a measurable delay from the camera picking up the light from ABD to the camera picking up the light from ACD, no?
On the other hand, if the one way speed of light IS instantaneous (and thus the bounced light travels slower), then there should be no measurable difference, and the light reflected off the mirrors should be detected by the camera from both mirrors at the same time.
When people explain the problem about measuring the one way speed of light, they often suggest that perhaps the speed of light in direction A is half the normal speed and in direction B the speed is instantaneous. That is done only to more easily demonstrate the problem. They aren't suggesting that this particular situation is likely. Perhaps light travels the same speed in both directions or maybe it's instantaneous in one direction and half the perceived speed in the other, but it could just as easily be
anywhere in between those two extremes. The point is that no matter how extreme the difference, you wouldn't be able to detect it because the information you are collecting is going to come to you no faster than whatever speed the light is traveling.
I'm not trying to measure a single path though.
The test is to see if there is a difference between two different paths that have the same source and endpoint.
If you'd be comparing the speed of light through any medium to its speed through a vacuum then the answer to that question is going to be "yes" - every time. Light CANNOT travel through ANY medium as quickly as it travels through a vacuum.
But that isn't relevant anyway because you can't know the one way speed of light through EITHER medium.
The issue here isn't with the set up. The issue is one of epistemology. No matter how clever the setup, you're still trapped inside the same epistemological box. The only way out is to assume the answer in order to test for it, which means you're not measuring anything at all. Our ability to know something is limited by our ability to collect information. Information has the same speed limit as light does and so, in the case of the one-way speed of light, our knowledge can't outrun the data collection, thus scientific epistemology is limited by the speed of causality in this physical universe.