Gauss-Markov theorem

Soledad Villar (NYU) and I discussed the Gauss–Markov theorem for the Nth time today. It's a strange theorem: It says that the best linear regressor in a regression problem is the one delivered by linear least squares. But it seems to conflate two kinds of linearity. It is both the case that the regressor must be linear in the features, and that the regression coefficients must be determined by a linear operation on the data. These are two different kinds of linearity I think! I'm confused about the assumptions that are required for this theorem; in particular it seems to be brittle when we generate the features by some non-trivial process in a latent space. I sure don't understand it yet!