Who was the first to discover that the curvature of an embedded surface is the product of the principal curvatures?

I don't know who originally defined Gauss curvature, but here is a long comment about the curvature of a surface. The second fundamental form of a surface is usually defined in terms of the Gauss map. It seems unlikely to me that curvature was originally defined in terms of the Gauss map, and I think Gauss introduced it because it is much easier to do computations using it.

Here is what I consider a more natural and maybe the original way to define the second fundamental form:

Given a point on a smooth surface, you can always move the surface so that the point is at the origin and the tangent plane is the $xy$-plane. The surface is therefore locally the graph of a function $f$ over the $xy$-plane.

The function $f$ is uniquely defined up to sign and a rotation of the $xy$-plane. Therefore, any function of $f$ and its derivatives, evaluated at $0$, that is invariant under sign and rotations defines a pointwise geometric invariant of the surface.

Since $f(0) = \partial_xf(0) = \partial_yf(0) = 0$, the simplest possible geometric invariant, up to a rotation of the $xy$-plane, is the Hessian, $\partial^2f(0)$. This defines the second fundamental form. The eigenvalues of $\partial^2f(0)$ are rotationally invariant and therefore define geometric invariants of $S$ up to sign. This defines the principal curvatures. Any even function of the eigenvalues defines a local geometric invariant, period . In particular, the determinant of $\partial^2f(0)$ is such an invariant and clearly a fundamental one. This defines Gauss curvature.

This provides an easy way to define local extrinsic geometric invariants of a surface. What is not at all obvious is how to identify which are intrinsic. If you try to do any computations using definition above, you get a horrible mess. I speculate that the real contributions of Gauss are introducing the Gauss map, to simplify calculations involving the second fundamental form and identifying Gauss curvature as an intrinsic invariant.

ADDED: The definition above of the second fundamental form can be used to give a straightforward proof for the Theorema Egregium of Gauss. I wrote a short note here: http://www.deaneyang.com/papers/gauss.pdf

Let me start with your first quotation:

"In 1763, Euler started a thorough study of curvature of embedded surfaces. In 1767, he found an expression of the curvature in terms of the product of principal curvatures."

Karin Reich's article in Leonhard Euler: Life, Work and Legacy (p. 482 in particular) explains what this is supposed to mean. As you know, Euler attacked the curvature of surfaces at some point P in terms of the curvatures of curves cut out by planes normal to the tangent plane at P. He found that the planes for which the maximal and minimal curvature occurs are orthogonal to each other, and then he developed a formula for the curvature of the curve cut out by any plane normal to the tangent plane in terms of the maximal and minimal curvature and the angle $\phi$ between the plane and the principal section. So "the curvature" here is definitely not Gauss's curvature.

The remark about the years 1763 and 1767 is even "curiouser": Euler presented his article E333 in 1763 to the Berlin Academy and it was published in 1767. So we are dealing with one and the same paper here.