In 1905, Einstein wrote in his relativity paper, "On the Electrodynamics of Moving Bodies": "The theory to be developed here is based, like all electrodynamics, on the kinematics of the rigid body, since the assertions of any such theory concerns with the relations among rigid bodies (coordinate systems), clocks, and electromagnetic processes".1If you had a truly rigid body, then you could communicate faster than light. Light takes a nanosecond to travel a foot. But if you push a rigid stick, and the other end moves immediately, then that is faster than a nanosecond. Relativity says that faster than light communication is impossible.
Einstein defined position by "means of rigid measuring rods and using the methods of Euclidean geometry".2
John Stachel explains that according to special relativity information cannot travel faster than the speed of light. Thus there can be no rigid body, which is possible in classical mechanics where forces are transferred at infinite speeds. A rigid body moves in a rigid manner, no matter what forces are imposed on the body. In fact, rigid motions can be defined without any contradiction in special relativity, even though a rigid body does not exist in the special theory of relativity.
Einstein spoke about rigid body because in 1905 he did not realize that this concept of the rigid body is incompatible with the special theory of relativity, and must be replaced by the concept of rigid motions.
(Update: Laue appears to have been the first to point out in 1911 that rigid bodies are not really rigid.)
She does not mention Poincare's approach, which had no such defects. He defined Poincaré–Einstein synchronization of clocks in papers during 1898-1904. Einstein's 1905 paper uses the same method, without crediting Poincare. Poincare's 1905 long paper avoids rigid bodies and defines lengths this way:
How do we perform our measurements? By transportation, one on the other, of objects regarded as invariable solids, one will answer immediately; but this is not true any more in the current theory, if the Lorentz contraction is admitted. In this theory, two equal lengths are, by definition, two lengths for which light takes the same time to traverse.These foundational aspects of relativity seem fairly trivial, but they are a large part of why Einstein is credited, because Lorentz did not say them. Lorentz later admitted that he had relativistic time as a mathematical trick, but did not have the synchronization method to relate it to actual clocks. This is the only part of relativity where Einstein's 1905 understanding can be said to be superior to Lorentz's. Since a rigid body like a meter stick is held together by electromagnetic forces, there are two explanations for the FitzGerald-Lorentz contraction. The first is that the motion distorts the fields so as to pull the molecules closer together, and the second is that the motion distorts space itself. FitzGerald and Lorentz offered the first explanation. Einstein did not offer any explanation in 1905, but did not express any disagreement with Lorentz. Poincare was the first to offer the second explanation, writing in 1905 that relativity is “common to all the physical phenomena, would be only apparent, something which would be due to our methods of measurement.”
But I don't know how Stachel or Weinstein or anyone else can recite this story without mentioning that Einstein's version is just a poor imitation of Poincare's.
Update: A comment points to Rigid body motion in special relativity for a modern explanation.