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 goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished).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>, 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.

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Because of Binet's independent work, some authors speak of the Gauss-Binet-Bonnet theorem, here is one <A HREF="https://tel.archives-ouvertes.fr/LASIE/hal-03363070v1">example.</A>

And here is the <A HREF="https://books.google.nl/books?id=04R7tOWnt2oC&pg=RA2-PA129">footnote</A> by Bonnet, in which he credits Binet.
 
[![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