In 1895, Lorentz published his proof of his "theorem of corresponding states". The purpose was to explain how Michelson observed the same speed of light in different frames of reference. He extended it to massive particles in 1899, and to velocities close to light in 1904.
To explain this, here is the notation. For a stationary frame, use position (x,y,z), time t, electric and magnetic fields (E,B). These satisfy Maxwell's equations.
Theorem. (Lorentz) For a moving frame, there exists a transformation of (x,y,z,t,E,B) into new variables (x',y',z',t',E',B') that satisfy Maxwell's equations and explain the Michelson-Morley experiment.
His statement is a little more complicated because he also considered charged particles, densities, and currents. The theory is called "corresponding states" because the Lorentz transformations define a correspondence between the stationary and moving frame variables. Einstein converted Lorentz's theorem into a postulate:
Postulate. (Einstein, 1905) Let (x',y',z',t',E',B') be the variables for a moving frame. Assume that they satisfy Maxwell's equations, just like (x,y,z,t,E,B).
From this, Einstein deduces that (x,y,z,t,E,B) and (x',y',z',t',E',B') correspond according to the definitions and formulas that Lorentz and Poincare found earlier.
Einstein calls this the principle of relativity, following Poincare's terminology without mentioning him. Einstein describes it as the same laws holding good in different frames. What he actually uses in his famous 1905 paper is the above postulate. That paper said:
They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.Without mentioning Lorentz, Einstein is referring to Lorentz's 1895 theorem of corresponding states, without recognizing the 1904 improvement to high velocities.
Poincare often stated the relativity principle as an observer being unable to detect uniform motion. He was alone in discovering a profound geometrical realization of it.
Theorem (Poincare, 1905) There is a non-Euclidean geometrical interpretation of (x,y,z,t) and Maxwell's equations such that the above transformations are simple consequences of the geometry.
A recent pro-Einstein book said, "But, of course, Einstein at first didn't completely understand the worldview that came from the special theory of relativity.". Einstein did not understand that non-Euclidean geometrical interpretation until several years later. Poincare published it first, then Minkowski got it from Poincare, and other leading physicists got it from Minkowski.
The difference here is that Lorentz and Poincare were giving new results and proving them. Einstein was just taking what Lorentz had proved, and restating it as a postulate. Lorentz said that Einstein simply postulates what we have deduced. Poincare's view became the modern view.
Proving scientific results based on previous theory and experiment is consistent with philosophical positivism. The philosophers and Einstein scholars say that Einstein was anti-positivist because he ignored the previous work and just postulated what he thought ought to be correct. But as you can see, he did not ignore the previous work. He accepted the conclusions while ignoring some of the reasoning behind them. And he did not reach the essence of the theory.