The question gets a little confusing when algebraic group language is added.   But on the level of finite groups of Lie type, it's useful to keep in mind that *Jordan decomposition* in the algebraic group sense has intrinsic meaning: given $p$, an element of a finite group decomposes uniquely as a product of two commuting elements, one of $p$-power order and the other of order prime to $p$.   This decomposition is preserved under any finite group homomorphism.

In any case, the initial question concerns only finite groups.  Only the case when the representation is nontrivial need be considered. For the given group $G =$ SL$_2(q)$, whose order is $q(q-1)(q+1)$, the matrices which are already diagonal have orders dividing $q-1$ (while other semisimple elements have orders dividing $q+1$ and can be diagonalized over a quadratic extension of $\mathbb{F}_q$).  On the other hand, $G$ is simple if $p=2$ and simple modulo a center of order 2 if $p$ is odd, while it has an element of order $q-1$.   It follows that unless the representtion is *trivial*, the field $k$ must have at least $q$ elements and a diagonal matrix must map to a matrix diagonalizable over $k$.