I'm going to give youy my definition of a free module.
Let $X$ be a non-empty set and $M$ be a $R$-module. We say $M$ is free on $X$ if for every map $f: X \to N$ ($N$ a $R$-module) there exits a unique $\phi \in Hom_R(M,N)$ such that $$\phi \circ \iota = f $$ Where, $\iota: X \to M$.
It's easy to see from the uniqueness of $\phi$ that $\iota$ must be injective, however I was not able to prove that $\iota(X)$ is a $R$-lineraly independent subset of $M$.
Note: $X$ is not necessarily a subset of $M$.
I'm going to treat $X$ as a subset of $M$ because it's easier. Suppose there is a nontrivial dependence relation $$\sum a_xx=0$$ Then there is no module homomorphism $$M\to\prod_{x\in X} R$$ such that the image of $X$ is independent. But there is a map from $X$ into the direct product whose image is independent.