How can I show that the speed of light in vacuum is the same in all reference frames?
For a basic treatment of the Michelson-Morley experiment please see 1. It's not important to know the technical details of the experiment to answer your questions though. The only relevant thing is the result, let me put it in basic terms since you seem to struggle with the "physics slang":
While the total velocity of a ball thrown from a truck is the sum of the velocity of the ball relative to the truck and the velocity of the truck relative to the observer, the velocity of a light beam emitted from the truck is not. Much more the velocity of the light beam seems completely independent of the velocity of the truck.
Michelson and Morely didn't have a truck, they had the earth orbiting the sun.
Please make it clear to yourself that this experimental fact can be explained by stating that the speed of light is constant. If I say to you the speed of light is constant in every frame of reference, then the above result isn't surprising at all to you.
But you want more. You want me to prove to you that the speed of light is universally constant. I cannot. There will never be an experiment that shows that this axiom is universally true. How should one ever construct such an experiment, how should one, for example, test the theory in the Andromeda galaxy? It's impossible, but it doesn't matter: Why not just stick with the axiom, as long as we can explain everything we see around us with it?
As you already said there's an interesting connection between the invariance of the speed of light and Maxwell's equations. One can indeed prove that the speed of light has to be constant, otherwise, Maxwell's theory can't be true for all inertial frames. But this is no proof that can convince you either, since accepting Maxwells equations is no different to accepting the invariance of the speed of light. Furthermore, the basis of Einstein's theory is not the invariance of the speed of light, but the invariance of the speed of action. Which cannot be concluded from Maxwell's theory, even though it's a reasonable guess.
Physical theories are not provable. But as long as they comply with reality, we accept them as truths.
Addendum: I recommend this short lecture for layman by R. Feynman on the topic. Feynman and I present a very similar line of reasoning.
It's hard to fully understand what you're asking but here are some things that might help:
The Michelson-Morley (MM) experiments don't show the speed of light is constant, it just rules out particular kinds of ether (the kind that can freely flow past particles). Ether is the supposed thing that light waves "oscillate" in.
You are right to say that's circular reasoning.
Maxwell's equations don't prove the speed of light is constant. But they suggest it if you also assume it's not possible to tell which frame you are in.
Einstein came to his conclusions based on gut instinct that electromagnetism had to obey the principle of relativity. He took this one step further and decided to elevate the idea to a principle and see what that lead to.
There is no proof the speed of light is constant except experiment - you can't do it theoretically. There are some arguments that came after Einstein, based on the idea of causality, etc.
Remember Einstein did physics by having convictions about the way the world worked and this worked exceptionally well for relativity but not for quantum non-locality.
That something was thought to be the aether, but in the absence of that why could it not be relative to whatever emitted it?
This seems to be the key point of your question.
Now, historically speaking this alternative had already been ruled out on theoretical grounds, because it isn't compatible with Maxwell's equations. However, since you are asking for experimental evidence, consider astronomical observations of binary star systems: it seems obvious that if the speed of the light coming from a star in the part of its orbit when it is approaching us were different to the speed of light coming from the same star in the part of its orbit when it is going away from us, it would cause observable effects.
In order to help quantify the extent to which making the speed of light relative to the speed of the light source would affect astronomical observations, I did a Google search on the phrase "astronomical observations binary star systems speed of light" which found this article by Tedd Bunn, chair of the department of physics at the University of Richmond. The answer is that the effect would be extremely obvious:
[...] you might wonder whether this effect would be significant for real star systems. The answer is that it turns out to be extremely significant. The reason is that the stars are very far away. That means that, even though the difference in the light's speed at the top and bottom of the orbit would be very slight, the faster light would still have lots of time to overtake the slower light. In fact, in real star systems, you'd end up seeing not just three or five images of the star, but thousands of images.
Needless to say, we don't see that sort of thing at all, and that provides extremely strong evidence that the speed of light does not depend on whether the source is moving towards you or away from you.