I'm not a specialist, but from browsing the french literature it appears that the interpretation mentionned is correct: Leray was motivated by studying general invariants of spaces and continuous functions, this from his interest in Mechanics and PDEs, see the quote page 6 of this (Leray was in fact chair of Mechanics at the Académie after WWII, and also chair of ODEs and Functional Equations at Collège de France). For instance in this document (warning: 60MB) are his publication list and some scanned notes from pre-WWII meetings with Bourbaki where Leray is listened to precisely about spectral theory matters. Also, Leray learned from Elie Cartan a lot about Lie groups and representation theory, and knew its relationship to quantum mechanics (i.e. again the idea of invariants).
The first paper of Leray on the topic of spectral sequences where really the word spectral appears is the comprehensive one published in 1950 (here is its Zentralblatt review), so the paper was circulating earlier. Apparently a first note in CRAS by Leray dates from 1945, then in 1947 Koszul generalized the idea, but still without the word spectral I think. These were treating cohomology stuff. On the other hand, Serre's CRAS note, which predates his thesis, appeared in 1950, and it treats homology stuff. For cohomology matters, I've seen in early papers anything from "Leray spectral sequence", to "Leray-Koszul", to "Leray-Koszul-Cartan" (since Cartan had a seminar on those things).
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