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 The formula for triangles goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished).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-Bonet theorem, here's one example.