Shiing-Shen Chern's <A HREF="https://doi.org/10.1142/9789812812834_0033">Historical remarks on Gauss-Bonnet</A> seems an authoritative source. The <A HREF="https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem#For_triangles">formula for triangles</A> goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished). The <A HREF="https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem#Statement">formulation for compact surfaces</A> was written up later by von Dyck (1888).

Some authors in fact refer to the Gauss-Binet-Bonnet theorem, here's one <A HREF="https://tel.archives-ouvertes.fr/LASIE/hal-03363070v1">example.</A>

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Here is the <A HREF="https://books.google.nl/books?id=04R7tOWnt2oC&pg=RA2-PA129">footnote</A> by Bonnet, in which he credits Binet with the independent derivation.
 
[![enter image description here][1]][1]

<sub>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.</sub>


  [1]: https://i.sstatic.net/VWcOd.png