St. Louis was teeming with activity a century after having gained fame as the starting point for the Lewis and Clark expedition in 1803. To commemorate the centennial of the Louisiana Purchase, the city hosted a World’s Fair that attracted almost twenty million visitors during its seven-month run. The 1904 Olympic Games were also held in St. Louis in conjunction with the Fair, ...Darrigol says in that 2004 paper:
Henri Poincaré, the final speaker, drew the largest audience. His lecture, “L’état actuel et l’avenir de la physique mathé-matique” , was trans-lated in two separate ver-sions: “The principles of mathematical physics” in the conference proceedings by G. B. Halsted  and “The present and the future of mathematical physics” in the Bulletin by J. W. Young . ...
Poincaré opened his remarks with four questions. “What is the present state of mathematical physics? What are its problems? What is its future? Is it about to change its orientation?” [27, p. 240]. Regarding these questions he stated, “It is easy to ask; difficult to answer” [Ibid.]. With Einstein’s annus mirabilis less than a year away, the answer to the third question was somewhat different from what Poincaré might have imagined, although his remarks at the St. Louis Congress attest to the fact that he came very close to discovering the theory of special relativity. Darrigol  provides a recent, balanced account of the controversy surrounding the discovery of special relativity.
To sum up, the similarities between Poincaré’s and Einstein’s theories of relativity can be explained in two different ways: by common circumstances or by direct borrowing. ...So either Einstein plagiarized Poincare, or Einstein independently reinvented some of Poincare's ideas. (We know Poincare did not get ideas from Einstein, because Poincare published first.)
Most trivially, a historian biased in favor of Poincaré’s precedence over Einstein will tend to favor the direct-borrowing thesis and a historian with the opposite bias will favor the common-circumstances thesis.
You cannot trust what Einstein said, because he hardly ever credited anyone for anything. A rare exception was crediting Poincare for E=mc2 in 1906. But for the most part, he refused to even comment on what Poincare did, so Einstein's story is not credible.
Regardless of that issue, the heart of special relativity is the spacetime geometry and electromagnetic covariance. Poincare published these in 1905, and Einstein did not even understand them until years later. There is no serious dispute about these points.
This is all explained in How Einstein Ruined Physics.
Another new paper by Stephen Boughn and Tony Rothman analyzes a pre-Einstein 1904 paper on mass-energy equivalence, and says:
On learning about Hasenoehrl's calculation, the first question from any physicist is in-evitably, "Did Einstein know about it?" There is, to our knowledge, no smoking gun that provides a permanent answer to this question. Einstein was not in the habit of citing others in his early papers, a lapse which has given rise to countless myths and specula-tions about what he knew and what he didn't. On the one hand it seems unlikely, if not incredible, that he would not have known of an award-winning paper that appeared in the Annalen der Physik, the leading physics journal of the time, to which he had himself already contributed five papers before 1905. On the other hand, he always insisted on his priority in the matter. In a 1907 paper Johannes Stark had credited Planck with E = mc2. Einstein replied testily, "I find it rather strange that you do not recognize my priority in the relationship of inertial mass and energy," to which Stark responded contritely that upon rereading Planck he realized that the latter's work was rooted in Einstein's own, at which Einstein himself apologized for his pettiness.See a pattern here? Einstein aggressively sought credit while ignoring previous work. Whether or not you think that this is a serious character flaw, you cannot trust what Einstein said.
Update: There is now an English translation of Hasenoehrl's 1904 paper.