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.