Question: Let $S\subset \mathbb{R}^3$ be a closed convex surface and let $p,q\in S$ be points such that $d(p,q)=\operatorname{diam}(S)$ where $\operatorname{diam}(S)$ is the diameter of $S$ with respect to the *intrinsic* distance $d$ of $S$. 

Does $d(p,q')=\operatorname{diam}(S)$ imply $q=q'$?

Answer: 
Take two copies of an equilateral triangle $ABC$ and glue them along their boundary. The resulting surface is homeomorphic to the sphere. The metric is convex. Then both $B$ and $C$ are at maximal distance from $A$ so $B$ is not the unique point at maximal distance from $A$.
Make two cuts from a cylinder along two planes that both contain a point $p$ on the surface. If the cuts form the same close to $\pi/2$ angle with the axis you get a surface bounded by a portion of the boundary of the cylinder and two ellipses. There are exactly $2$ points $q$, $q'$ so that $d(p,q) = d(p,q')$ is the diameter of the domain. This would be an oddly shaped salami cut.