Local time measured by clocks

To a lot of people, the essence of relativity is the Lorentz transformations, or the spacetime geometry. Since Einstein did not have anything to do with discoverying those, the historians who credit Einstein usually point to his superior understanding of time. Einstein himself tried to claim credit that way, and wrote in 1907:

Surprisingly, however, it turned out that a sufficiently sharpened conception of time was all that was needed to overcome the difficulty discussed. One had only to realize that an auxiliary quantity introduced by H. A. Lorentz, and named by him 'local time', could be defined as 'time' in general. If one adheres to this definition of time, the basic equations of Lorentz's, theory correspond to the principle of relativity ...

Poincare had that realization before Einstein, even if Lorentz did not.

Mario Bacelar Valente writes in a new relativity paper:

In a way this change in the metrology of time was antecipated in developments in theoretical physics. In the late 19th and early 20th century several thinkers were involved with issues related to the so-called electrodynamics of moving bodies. In his criticism of Lorentz's electron theory and its extension to the case of matter in (inertial) motion, Poincaré noticed that what for Lorentz was a mathematical artifice – that of rewriting his equations for the case of moving bodies in terms of auxiliary variables one of which Lorentz had called local time –, could have a completely different physical interpretation. According to Poincaré the local time can be the time being measured by observers in motion with the material bodies in question:

I suppose that observers placed in different points set their watches by means of optical signals; that they try to correct these signals by the transmission time, but that, ignoring their translatory motion and thus believing that the signal travel at the same speed in both directions, they content themselves with crossing the observations, by sending on signal from A to B, then another from B to A. The local time t' is the time indicated by watches set in this manner. (Poincaré 1900; cited in Darrigol 2003, 359)

This procedure for synchronizing clocks had been presented by Poincaré, for the particular case of clocks taken to be at rest, in an earlier work published in 1898. In this work Poincaré mentions that even the best clocks, by that time still mechanical clocks, had to be calibrated to the sidereal time:

In fact, the best clocks must be adjusted from timeto time, and these adjustments are made with the help of astronomical observations; arrangements are made so that the sidereal clock marks the same hour when the same star passes over the meridian. In other words, it is the sidereal day, that is the duration of the Earth's rotation, which is the constant unit of time. (Poincaré 1898, 3; my translation)

We can still read Poincaré’s 1900 remarks in the light of his 1898 memoir, i.e. implicitly, the clocks are taken to have been calibrated to sidereal time. However, due to the issue of the setting of the initial phase of distant clocks (the synchronization of the clocks), Poincaré discusses the relation between the time read by the clocks in an apparently autonomous way.

That is correct. Einstein first wrote about relativistic time in 1905, long after Poincare's ideas were well-known throughout Europe.