Is Gaussian curvature intrinsic in higher dimensions?

To answer the question you asked in the comments: Yes, the Gaussian curvature of a hypersurface is an intrinsic isometry invariant up to sign. One reference for this is Volume 4 of Spivak's Comprehensive Introduction to Differential Geometry. In my second edition it's Corollary 23 in Chapter 7.

It is intrinsic for $n$ odd, and not if $n$ is even. If you pick a normal field $N$, you will compute the Gaussian curvature (Gauss-Kronecker curvature) $K = \det S$, but if you picked the other normal field $-N$ instead, you would have the shape operator $-S$ associated and $\det S = \det(-S)$ only if $n-1$ is even, that is, if $n$ is odd. The curvature shouldn't depend on the choice of normal field if it is to be intrinsic.