NO. That is not correct. In fact, it's catastrophically wrong.
The experiment was to address the question of how light propagates. Specifically, do light waves propagate through a medium, similar to sound waves propagating through air, water, wood, etc? A leading theory at the time was that light propagated through a medium called the "luminiferous aether". The null result, that is that there is no difference in the speed of light regardless of the angle between Earth's direction of motion and the light's direction of motion, shows that there in fact is no "aether" and the light does not require a medium for propagation. Earth's motion was already clearly demonstrated through astronomy.
And mass-energy equivalence has nothing to do with it.
So, if you swim with the current of the river, you’re not swimming faster than against the current? Light beams shot through the anther in the direction we are moving and light beams shot perpendicular, should bounce back at the same speed? There should’ve been a difference.