Shiing-Shen Chern's Historical remarks on Gauss-Bonnet seems an authoritative source. The formula goes back to Gauss (1827), Bonnet (1848), and Binet (unpublished). Gauss considered a triangle, Bonnet and Binet generalized it to smooth
closed curves, where the sum of the angles is replaced by the integral of the geodesic curvature.

The equation for compact surfaces of arbitrary genus, referred to by Needham, was written up later by von Dyck (1888) (relying on earlier developments by Kronecker). So yes, the question in the OP "is Needham accurate" can be answered in the affirmative.

Daniel Gottlieb has examined the "sociology of mathematics" which governs the naming of theorems. In the context of the Gauss-Bonnet theorem he writes: *Part of this story shows that the name of a theorem is not really for an
attribution. It is very convenient to have a name for important theorems, and the
main point is that people should know approximately what theorem is meant by
the name rather than who gets the credit. Still, one can reflect that Bonnet's name
is famous and Dyck's is virtually unknown these days.*

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.}

geometry. However, in modern differentialtopologybooks it is usually stated as a relation between the Euler characteristic of a compact surface and the integral of the curvature over it. Perhaps this distinction is at play. $\endgroup$