Evaluating correctness of various definitions countable sets
The Wikipedia theorem is correct. More generally, if $A$ is an ordered set and there exists a surjection $g\colon A \twoheadrightarrow B$ then there exists an injection $f\colon B \hookrightarrow A$ given by $f(b) = \min g^{-1}(b)$. This is an injection since $g(f(b)) = b$.

More generally, if you assume the axiom of choice then you don't need to assume that $A$ is ordered, since the axiom of choice guarantees the existence of a function $f\colon B \to A$ satisfying $f(b) \in g^{-1}(b)$ for all $b \in B$.