New Definition of Convex Hull

This is in a locally convex topological vector space over $\mathbb R$. The equivalence is for the closed convex hull, using closed half-spaces, closed convex sets, and continuous linear functionals. That is, the closed convex hull of $X$ is the intersection of all closed convex sets containing $X$, and this is also the intersection of all closed half-spaces containing $X$.

Counterexample to your statement: In $\mathbb R^2$, let $X$ be the union of the open half-space $\{(x,y): x > 0\}$ and the point $(0,0)$. This is a convex set; its convex hull is itself. But any half-space containing $X$ contains the whole $y$ axis, so the intersection of these half-spaces is $\{(x,y): x \ge 0\}$.

A closed half-space is a closed convex set, and the intersection of closed convex sets is a closed convex set, so the intersection of all half-spaces containing $X$ is a closed convex set containing $X$.

Conversely, if $C$ is a closed convex set and $p$ a point not in $C$, the Hahn-Banach separation theorem says there is a closed half-space containing $C$ but not $p$. So any point in the intersection of all closed half-spaces containing $X$ is in the closed convex hull of $X$.