Let $A$ be a ring, $E$ a right $A$-module and $F$ a left $A$-module. Consider the free $\mathbf{Z}$-module $\mathbf{Z}^{(E\times F)}$ which comes with the injective canonical mapping $\phi:E\times F\rightarrow\mathbf{Z}^{(E\times F)},\,(x,y)\mapsto e_{x,y}$, where $e_{x,y}:=(\delta_{(x,y),z})_{z\in E\times F}$ for $(x,y)\in E\times F$.
Bourbaki defines the tensor product of $E$ and $F$ as the quotient $\mathbf{Z}$-module $(\mathbf{Z}^{(E\times F)})/C$, where $C$ is the submodule of $\mathbf{Z}^{(E\times F)}$ generated by the elements of the form $(e_{x_1+x_2,y}-e_{x_1,y}-e_{x_2,y})$, $(e_{x,y_1+y_2}-e_{x,y_1}-e_{x,y_2})$ and $e_{x\lambda,y}-e_{x,\lambda y}$ for $x,x_1,x_2\in E$ and $y,y_1,y_2\in F$ and $\lambda\in L$.
Elsewhere, I have seen the element of the form $ne_{x,y}-e_{xn,y}$, with $x\in E$, $y\in F$ and $n\in\mathbf{Z}$, added to the list above. Is this necessary? Why does Bourbaki leave it out?
It's indeed not necessary, as these elements will already be in $C$ even for Bourbaki's definition.
Specifically, for $n\ge 1$, use induction to see it (let $x_1=nx$ and $x_2=x$ in the induction step).
For $n\le 0$, use the rule $e_{x\lambda, \, y} - e_{x,\, \lambda y} \in C$ with $\lambda=0$ and $\lambda=-1$.