I would define Stiefel-Whitney classes as the pullbacks of generators of $H^\*(BO, \mathbb{Z}/2)$ under a classifying map, and I gather this is a pretty common definition. 


However, the book "Characteristic classes" by Milnor-Stasheff contains a different definition, as ``eigenvalues'' for the Steenrod squares acting on the fundamental class of a manifold in its normal bundle. I want to attribute this construction to somebody for a paper, but I'm not sure what the correct attribution is. Who is the discoverer of this definition?



Who discovered this definition of Stiefel-Whitney classes?
For the relation $Sq(U) = \Phi(w)$, where $Sq$ is the total Steenrod squaring operation, $U$ is the Thom class, $\Phi$ is the Thom isomorphism and $w$ is the total Stiefel-Whitney class, I would cite Rene Thom's 1951 thesis, published as "Espaces fibres en spheres et carres de Steenrod" in Ann. Sci. Ecole Norm. Sup. (3) 69 (1952). In the introduction he explains that he found the results for manifolds embedded in Euclidean space, and that Henri Cartan then showed him how to obtain the general case. These results were first announced by Thom in two Comptes Rendus Acad. Sc. notes (t. 230, 1950, p. 427 and p. 507). Wen-Tsun Wu's formula $Sq(v)=w$ in the case of the tangent bundle of a closed manifold is not the same result. It was announced in the same volume of Comptes Rendus (p. 508 and p. 918). Its proof depends on the results of Thom.


