Validating a sequence of points as a convex hull
The answer is easier than you think. As you remarked you should check if you take always CCW or CW turns. But of course, this might result in a curve that is self-intersecting. But you can detect this easily, just sum up then angles of the "turns". Since your curve is closed this angle sum is a multiple of $2\pi$. If it winds up exactly once it does not self-intersect. In this case the angle sum is $2\pi$. Thus you are left with checking, if the angles sum up to $2\pi$.

See this paper for analogues in higher dimension.

Interestingly, in the smooth version (CCW turn is here positive sectional curvature), for a 1d curve you still have to check the winding number, but by an old theorem of Hadamard,  if higher dimensional (immersed) manifolds have positive sectional curvature everywhere, then they enclose a convex body.