The Michaelson Morley experiment is a static experiment in time, the interferometry isn't sensitive to when the light bounces off the mirrors, only to the distance to the mirrors. This means it doesn't serve to test the failure of simultaneity aspects of relativity, or the time-dilation aspects, It just measures spatial distances. So keeping the idea of a preferred frame, you can explain the null-result by saying the arm in the direction of motion contracted when moving relative to an absolute standard of rest, this is Fitzgerald theory.

Lorentz makes this contraction a natural thing by finding the transformations which keep the Maxwell-equations invariant, and then you get failure of simultaneity too, meaning moving observers have a tilted t axis and x axis, not a scaled up x axis (so shrunk down length measurements). The blown-up x-axis in relativity comes from the geometrical projection of a fixed relativistic-length moving arm to nonmoving coordinates, the geometry is explained from various points of view on stackexchange: http://physics.stackexchange.com/questions/14362/help-me-gain-an-intuitive-understanding-of-lorentz-contraction .

Really, to make relativity, all you need is to keep the Minkowski metric invariant, or equivalently, the wave-equation invariant, or in more elementary terms the speed-of-light invariant, this is Einstein's point. Then the physics becomes completely frame invariant, this is Poincare's point, and you reproduce the Fitzgerald contraction when analyzing a static experiment in a moving frame.

The distinction between the two is that Fitzgerald's idea is phenomenological, it just introduces an ad-hoc contraction, while the Lorentz theory is an actual theory that reproduces the phenomenology, with a nontrivial prediction regarding time dilation and so on. Michaelson-Morley as a test isn't strong enough to discriminate between the two possibilities experimentally.

as an experimental point, to discriminate between ether and Fitzgerald contractions and actual Lorentz transformation, you need an experiment which tests time, kinematics, or tests them indirectly using mass-energy or the dynamics of fast-moving electrons. Fast-moving muon lifetime experiments can directly test time dilation, but these came decades later. The experiment which actually was most sensitive to the time-aspects of relativity was the earliest of all, the 1851 Fizeau experiment. This did a velocity addition of a nonrelativistic velocity and the relativistic speed of light in a medium, and velocity addition is a real test of relativistic kinematics, it doesn't work in Fitzgerald theory (where time is unaltered, and the moving medium has negligible contraction, so velocities still should add normally). The Fizeau experiment was the main non-thought actual experiment which motivated Einstein, not Michaelson Morley, although Michaelson Morley motivated everyone else.

From Wikipedia's article on Fizeau's experiment, it seems that Lorentz introduced the failure of simultaneity to account for the results of the Fizeau experiment--- he used the correct first order in v relativistic tilt in the x-axis in the moving water medium, so the "t'=t-vx" part of the Lorentz transformation came directly from Fizeau. The denominator, the $\sqrt{1-v^2}$, is second leading order in v, and that came from Fitzgerald, and putting the two ideas together self-consistently, Lorentz made the transformations. The systematic modern geometric derivation seems to be due to Einstein and Poincare.

Historically, the first real test that discriminated between different dynamical theories of fast-motion was the change in the mass of a fast moving electron, detected by the shift in cyclotron frequency. This eventually came out on the side of Einstein, as opposed to an Abraham ether theory with a spherical electron. I don't think anyone actually bothered to make a pure Fitzgerald contraction theory, historically, possibly because relativity was obviously a better theory.

Theoretically, once you establish that the physics is fully frame invariant, you can throw out the ether, because the ether, by historical definition, was introduced as the thing that picked out the rest-frame. Once there is no rest-frame, any ether idea is just philosophy. The philosophy is discussed by Bell, in the one somewhat off-topic section of his book "Speakable and Unspeakable in Quantum Mechanics". It is not completely off-topic, because to reproduce quantum-nonlocality with hidden variables, you need nonlocal things to happen, which leads you to consider whether you can have a "hidden ether" which only serves to define some sort of objective wave-function collapse, as in Bohm's theory.