The first fundamental form of a surface $S$ at a point $p$ is "the quadratic form on the tangent plane $S_p$ inherited from the inner product structure of $\mathbb R^3$".
At the same time, the first fundamental form is apparently supposed to describe the surface at $p$ in some way, for instance you can compute Gaussian curvature from it. In particular, the first fundamental form should be different for different surfaces $S$.
So what is the first fundamental form?
It can't be the actual quadratic form $\langle x, x\rangle$ on $S_p$, that is, it can't simply be a function from $\mathbb R^3$ into $\mathbb R$, because that's the same for any surface $S$ that has the same tangent plane at $p$.
It can't be the triplet of coefficients $(E, F, G)$ either, since they depend on the parameterization used, and in fact I think by using the right parameterization we can get any triplet $(E, F, G)$ we want.