Stecklov operator vs Laplacian operator
I'm not entirely sure, but I guess it might have to do with the notions of intrinsic vs extrinsic curvature. The Laplace–Beltrami operator on a manifold is, I think, intrinsic in the sense that it can be constructed from the "internal" geometry within the manifold itself, and doesn't depend on any embedding in an ambient space. The Steklov operator, on the other hand, looks like it's written explicitly in terms of an embedding, so would be extrinsic in that sense.