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.
This has absolutely nothing to do with relativity. Of course A says that his own clock ticks at one second per second, as does B. This is as true in Newton's world (or in Aristotle's) as in Einstein's. The key question for relativity is: How fast does A say that B's clock ticks? And your analysis does not address this at all.
Best Answer
You find this hard to refute because your friend is correct in one sense: the MM experiment did not prove Einstein's second postulate of relativity, namely that the speed of light is constant for all inertial observers.
Recall that the Michelson-Morley experiment was designed to detect motion relative to an aether, or material medium for light. If your experiment on an open train carriage measured the speed of sound, then you would indeed measure different speeds along and across the carriage. So the MM experiment cast serious doubt on the notion of an aether.
Now, it was well known that Maxwell's equations did not keep their form under Galilean transformations between inertial frames. This was thought to be fine because the notion of a medium for light was believed before the MM experiment, so that the wave equation for light should transform in the same way as the wave equation for sound between inertial frames.
So along comes Einstein and says, given there's no medium, let's see what happens to our physics if we assume that Maxwell's equations keep their form under a transformation between inertial frames. He postulated therefore that the speed of light would be measured to be the same for all inertial observers and concluded that (1) the transformation group was the Lorentz, not the Galilean group and (2) the time measured between two events would in general depend on the observer. (1) was already know at the time of Einstein's 1905 paper, (2) was radical.
So the MM experiment motivated the assumed Lorentz covariance of Maxwell's equations and thus the new relativity postulate that the speed of light would be measured to be the same by all inertial observers.
The second relativity postulate therefore comes into play only when we compare the light speed measured by different inertial observers. Someone observing a light source on your train would notice a very different transformation law from the approximate Galilean transformation law that would describe the ping-pong ball velocity transformation to an excellent approximation.