In linear programming I ve seen two versions of the fundamental theorem, and I wonder whether there is a correspondance between them, so that given one version you don't have to prove the other one. The two versions I am talking about can be found in the links:
Also is it really necessary for the matrix to be full row rank? How is this assumption used in each proof?
Yes they are two different versions of the same theorem. The first one is the so called algebraic version, because it refers to the feasible basic solutions of a system of $m$ equations. The second one is the geometric version, because it refers to the extreme points of a convex set. You do not need to prove the counterpart, given one version, because the following holds
Theorem: $x\in \mathbb{R}^n$ is an extrem point of $P=\{x \in \mathbb{R}^n \ |\ Ax=b, x\geq 0\}$ if an only if $x$ is a basic solution of $Ax=b$ and $x\geq 0$.
Note that the Wikipedia geometric version is stated for bounded polyhedra (polytopes), but it still holds for unbounded polyhedra.
As far as it concerns the rank of matrix $A$, the full-rank hypothesis is used in the case 1 of the proof of the algebraic version (following the link in your question)
