This is a fascinating issue. And I don't think the "one idea by many minds" captures the core. On the one side, yes, there are ideas floating around, and most ideas will be in multiple minds. So there must be some truth to it. But: My experience tells me that it is often the details that matter. To the point: as John Carlos Baez (npub1nf4…nqe4) writes, Poincare had the math, but he was trapped in his metaphysics. And this is the key to the question. Einstein took relativity as an empirical operational fact and from this derived SR. Poincare, on the other hand, wanted to explain relativity from sth more fundamental. So he was rejecting that Einsteins axiomatic reasoning explains anything and he rather focused on the contracting electron as explanation for relativity. Here I would argue two points:
1. Poincare fooled himself, because he had to believe the contracting electron story in order to explain relativity thus assuming the premise (since the contracting electron story only was selected BECAUSE it had the right predictions). In a sense, he derived the maths, but being caught in the microscopic picture he was not able to distill the structural essence of relativity. Or with [Adlam] "it seems inaccurate to say that he had any significant intimations of special relativity before Einstein’s 1905 papers".
2. Einstein in contrast was transparent about the assumptions and in fact introduced a new way for physics: Operational axioms + the maths ==> A principle that later lead Heisenberg to Matrix mechanics and where Einstein took what was Poincare's seat.
[Adlam] https://arxiv.org/pdf/1112.3175