The problem addressed by Lorentz and subsequently Einstein was the speed of light. This emerged as a constant in Maxwell's equations, but if as was generally supposed, light is wave-like, it seemed reasonable to assume that it must be carried by some medium (the "luminiferous aether") at a velocity characteristic of that medium. Thus its velocity relative to an observer should have varied with the motion of the observer through the medium. Experiments of increasing sophistication failed to reveal any trace of that variation.This is mostly correct. Lorentz's program was more ambitious because he tried to deduce relativity from experiment and accepted principles, and because he looked for physical explanations. Einstein later described the importance of the Michelson-Morley experiment and how he looked for a more constructive theory, but acknowledged that he failed to get these for his famous 1905 paper. Little did he realize that his failure would someday be regarded as a sign of great genius!
Several explanations were put forward. It was proposed that the Earth must carry the local aether with it, but a more fruitful suggestion made independently by Fitzgerald (10) and Lorentz (11) was that objects moving through the aether must be somehow shortened along their direction of travel, thereby disguising relative changes in the velocity of light. It was supposed that intermolecular forces must be transmitted at the same velocity as electromagnetic waves, so that movement through the aether would influence the degree of attraction between molecules and thus the separation of those molecules. To effect a reconciliation with Maxwell's equations, it was necessary to assume changes not only of length, but also of time, and thus the LT, ...
The LT was already reasonably well known by 1905. There had been significant contributions to its development, not only from Lorentz and Fitzgerald, but also by (among others) Heaviside, Larmor and Poincaré. It was Heaviside's analysis of the disposition of fields accompanying a charged particle (the "Heaviside ellipsoid") that had suggested to FitzGerald the idea of length contraction (12). Larmor had described an early form of the LT and discussed the necessity of time dilation (13). Poincaré had recognized the relativity of simultaneity and had studied the group theoretic properties that form the basis for the covariance of the transformation (14). ...
Lorentz commented, presumably with some chagrin, that,Einstein simply postulates what we have deduced, with some difficulty, and not altogether satisfactorily, from the fundamental equations of the electromagnetic field (17).In what follows, the distinction drawn will be between Einstein's SR (ESR) and what we will call Lorentzian SR (LSR). This is not to diminish the contributions of others, but it was Lorentz in particular who sought to explain SR from underlying physical processes, as will be the objective below. Once the form of the LT was known, all else in SR then followed, including the composition of velocities, the group theoretic properties of the transformation, and the invariance of Maxwell's equations. It may be argued that with these refinements (largely due to Einstein and Poincaré), ESR and LSR are essentially equivalent. They cannot be distinguished, mathematically or empirically, through the privileged frame that was supposed by Lorentz, but declared "superfluous" by Einstein (1). It would seem that any such frame is rendered undetectable by the covariance of the LT. Nor can ESR and LSR be distinguished by supposing that in ESR, though not in LSR, the LT describes a transformation of spacetime. As we have seen, the LT must be explained in either case by changes occurring in matter as it is accelerated from one inertial frame to another.
Poincare (and later Minkowski) showed the covariance of the LT, and hence the equivalence of frames. Einstein also tried to apply Poincare's relativity principle, but only showed a correspondence of frames in the same sense that Lorentz had previously proved.
It is completely legitimate to say that the FitzGerald contraction of a solid object is caused by a distortion in the electromagnetic fields of the molecules, and that the distortion is predicted by Maxwells equations for motion thru a suitable coordinate frame. If you calculate the electromagnetic fields that hold the atoms and molecules together, apply the quantum field theory to get the atomic distances, and redo the calculations with the distortions from motion, then you will find that solid objects are contracted according to the FitzGerald contraction formula.
The modern view is that the contraction is a byproduct of the non-Euclidean geometry of spacetime developed by Poincare and Minkowski, and adopted several years later by Einstein. It is often useful to have more than one view in physics. Both views are completely correct and based on accepted physics.
I am not sure that Lorentz ever said that any frame was privileged. His theory was that the laws of electromagnetism would be valid in any frame. Einstein said more or less the same thing. The key step in showing that the theory did not need a privileged frame was Poincare's 1905 proof that the Lorentz transformations formed a group, and then that the group is a symmetry group for the theory. Lorentz and Einstein both missed this crucial concept.