Thanks for answering this question. I suppose the problem is more interesting if we consider, in your 2nd example, that the two (isomorphic) posets are different subposets of the lattice $L$. Can we characterize affine independence of $V$ in $\mathbb{R}^n$ by a property of the subposet $(V,\preceq)$ of the lattice $L$? Consider, for instance, a simple necessary condition: If there exist $U \subseteq V$ and $v \in V \setminus U$ such that $\sup U = v$ and $\forall u,u' \in U: \inf(u,u') = \emptyset$ then $U$ is lin. dependent because $\sum_{u \in U} u = v$. Is there an if-and-only-if-condition?

Is the question as trivial if we consider affine independence instead of linear independence?