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 The formula for triangles goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished).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, which is the equation referred to by Needham, was written up later by von Dyck (1888). So yes, the question in the OP "is Needham accurate" can be answered in the affirmative.
Because of Binet's independent work, some authors speak of the Gauss-Binet-Bonnet theorem, here is one example.
And here is the footnote by Bonnet, in which he credits Binet.
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.