To settle this:
As mentioned here, "order of contact" is related to the concept of tangency. For instance, we say that a curve and its tangent line have first-order contact, since the function and first derivative values agree at the point of intersection.
For the functions given in the OP, we know the Maclaurin ($x=0$) expansion $\sin\,x=x-\dfrac{x^3}{3!}+\cdots$; comparing this with $x^2$ shows that at $x=0$, the function values of these two functions agree, but the first derivative values don't. We can thus say that the order of contact is zero.
Put another way: it doesn't matter if the higher derivatives match; the order of contact between $f(x)$ and $g(x)$ is related to the degree of the first nonzero term of $f(x)-g(x)$. The first nonzero term of $\sin\,x-x^2$ is $x$, which is of degree $1$; the order of contact is thus $1-1=0$.