Clarification about the definition of free module I am reading this notes.
Definition 1:

Let $R$ be a commutative ring with $1$. Let $S$ be a set. A free $R$-module $M$
on generators $S$ is an $R$-module $M$ and a set map $i:S\rightarrow M$ such that,
for any $R$-module $N$ and any set map $f : S \rightarrow N$, there is a unique
$R$-module homomorphism $\bar{f} : M \rightarrow N$ such that $\bar{f}\circ i = f : S \rightarrow N$. The elements of $i(S)$ in $M$ are an $R$-basis for $M.$

Definition 2: $M$ is a free $R$-module if $M$ has a basis.
We know that Definition 1 is equivalent to Definition 2. So we can choose anyone of them to define free module. If we choose the Definition 2 then it is very easy to give examples of free module, for example, the $R$-módule $_{R}R$ is free, but we have to work a lot more in order to show that free modules really exist by using the Definition 1.
My question is: Is there a practical or theoretical advantage to choose the Definition 1 instead of the Definition 2?
PS I am self-studying Module Theory, and I find more natural the Definition 1.
 A: In practice one often views the map $i$ (as in definition $1$) as an inclusion; then definition $1$ defines a free $R$-module $M$ in terms of how you can define $R$-linear maps from $M$ to another $R$-module: just assign the values on $S$ as you wish, and you can linearly extend in a unique way. This is used all the time. Definition $1$ also makes the connection between the terms "free $R$-module", "free commutative $R$-algebra" (aka polynomial ring over $R$), "free group", etc. very clear: it basically the same universal property. This leads to the categorical point of view, e.g., that the functor $F:\mathbf{Set}\rightarrow\mathbf{R-Mod}$ (assigning to a set $S$ the free $R$-module $F(S)$ generated by $S$) is left adjoint to the forgetful functor $U:\mathbf{R-Mod}\rightarrow\mathbf{Set}$.
Also, the first definition is often used to define important objects in various branches of mathematics, for instance the singular chain groups. Let $X$ be a topological space and $S$ be the set of all continuous maps from the standard $n$-simplex $\Delta_n$ to $X$. Then the free $\mathbf{Z}$-module generated by $S$ is by definition the abelian group of singular $n$-chains of $X$ (denoted $S_n(X)$).
