What’s is the 1st Fundamental form? The coefficients or the dot product? I often hear that the coefficients of the 1st FF are the 1st FF. But the I also hear that the dot product is the 1st FF. But make sense to me but...which one is which? If the dot product is the 1st FF, then how come people say that Gaussian curvature can be written in terms of the 1st FF? This isn’t true since it cannot. It can be written in terms of the coefficients of the 1st FF. Is this just a colloquialism then?
 A: Given a surface $S \subseteq \mathbb{R}^3$, the first fundamental form is the inner product on the tangent space (at each point) on the surface. The first fundamental form doesn't have "coefficients" by itself. Once you choose some parametrization which covers some part of the surface, you get an induced basis for the tangent space and then you can represent the first fundamental form by a $2 \times 2$ matrix which has coefficients (the "famous" $E,F,G$). Different parametrizations (even if they cover the same part of the surface) will give you different matrices and different coefficients. 
If you know the parametrization and the matrix representing the first fundamental form, you can reconstruct the first fundamental form itself on the image of the parametrization so sometimes people identify the first fundamental form (restricted to the image) with its coefficients in some (implicitly understood) parametrization.
Finally, your question about the curvature is a good one. The easiest way to introduce the curvature is to do it locally via the coefficients and not via the inner product itself (which is a global thing defined on the whole surface). Then it is not so clear why the curvature depends on the inner product and not on the coefficients. However, to show that the curvature is a notion which is defined globally on the whole surface, one actually needs to verify that it is given by the "same" formula when you use different parametrizations and doing so implicitly shows that the curvature actually depends on the inner product.
More sophisticated approaches derive the curvature globally directly from the inner product (via the covariant derivative which by itself is defined in terms of the inner product) and so make it clear that the curvature can be constructed from the inner product without even talking about the coefficients.
