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 goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished).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-Bonet theorem, here's one <A HREF="https://tel.archives-ouvertes.fr/LASIE/hal-03363070v1">example.</A>