Why did Michelson believe the aether was moving? If a wave moves east towards California the wave-motion would be slower north-south. Is this the kind of way they were thinking about the aether as medium for light, of what was later, by Lorentz, styled a "dilation" (in the north-south motion)?  
 A: 
Why did Michelson believe the aether was moving?

He didn't, he believed the Earth was moving. And the Sun, and maybe other things.
In contrast, it was already understood that the aether was "universal", not necessarily moving or not moving on it's own, but whatever motion it had, it had to be the same motion everywhere.
Why? Well let's say the aether was moving north on one half of the universe and south on the other; in this case the light from different parts of the sky would have different frequencies due to the Doppler effect. And they definitely weren't seeing that, quite the opposite, the frequencies were so constant that they were already used to determine the chemical makeup of different stars through spectroscopy.
So if the aether is universal, then we have a great yardstick. It doesn't really matter if it's moving or now, we can just express our velocity in those terms, which is the whole idea of the Galilean transformation - speeds are relative. And since it's universal, we can use that to compare motion of different bodies.
Now of course the easiest one to start with is the Earth, because that's where we keep all our stuff. And we're pretty sure the Earth is orbiting around the Sun, just as sure as we are that the aether isn't orbiting around the Sun. So there should be a relative velocity between the Earth and aether, it should vary depending on the time of day, and the exact magnitude should vary by season.
And yet none of this happened. So then things got complicated. If the Earth isn't moving compared to the aether, then what...


*

*the Earth just isn't moving

*the aether is dragged around by the Earth

*the aether doesn't exist


Option 1 was eliminated immediately, we're not that special. Option 2 resulted in lots of new work, but it all ultimately failed. And so... option 3.
A: Michelson and most physicists of the time thought that the (A)ether was moving relative to the Earth, or, amounting to the same thing, that the Earth was moving relative to the Ether. They had taken on board, from Copernicus, Galileo and their successors, that it was better to think of the Earth moving round the Sun, and not the other way about. So if the Ether was a universal medium permeating the solar system and beyond, it was natural to think of the Earth as moving through it.
A: Michelson postulated that the earth and the ether were moving at different velocities; one was moving relative to the other.  The first scenario had the earth move through and ether, while the second hand an ether wind blowing over earth.  Nevertheless, he ultimately proved that there was no such ether.  Since light reached the same point at the same time even though one travelled vertically and the other horizontally.
A: I believe there is a small misunderstanding as to what was moving.
At the time of Michelson and Morley's experiment, it was postulated that an aether must exist for light to be able to propagate.  It was also known that the earth revolves on its axis and moves around the sun.
Since the experiment's apparatus was "attached" to the earth, its motion trough the aether, would be equivalent to the eather moving through the apparatus.
In other words, by making the apparatus the "frame of reference," it gives the appearance that the aether is the one moving!
