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?