I understand that for gamedev it's more natural to think in terms of the exterior product of two vectors rather than the normal, because you have to deal with t

I understand that for gamedev it's more natural to think in terms of the exterior product of two vectors rather than the normal, because you have to deal with transformations a lot. But the normal is a vector, it's a definition that has been there for hundreds of years. There is no need to rename stuff, just use the thing that better fits your use case. In fact there are other things that are easier to understand in terms of normals, like the shape operator of a surface.