Shiing-Shen Chern's Historical remarks on Gauss-Bonnet seems an authoritative source. The formula for triangles goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished). The formulation for compact surfaces was written up later by von Dyck (1888).
Some authors in fact refer to the Gauss-Binet-Bonnet theorem, here's one example.
Here is the footnote by Bonnet, in which he credits Binet with the independent derivation.
After having completed this paper, I saw a note by Binet, appended to a paper by Olinde Rodrigues in the "Correspondence of the École Polytechnique". In that note Binet derived Gauss's theorem in a similar way as I did.