Definition of a riemannian metric (smoothness)

In Ana Cannas de silva's book Lectures on Symplectic geometry he defines a positive inner product to be smooth when for any vector field $v$ the function $x \mapsto g_x(v_x, v_x)$ is smooth.

Some other literature (i.e my lecture notes and wikipedia) require that for any two different vector fields $v,w$ we have that $x \mapsto g_x(v_x, w_x)$ is a smooth maps.

Are these two conditions equivalent? If so, how?

Hint: If $b(-,-)$ is a symmetric bilinear form in characteristic $\neq 2$, then $$b(x,y) = \frac{1}{2}\left(b(x+y,x+y) - b(x,x) - b(y,y)\right).$$