To be specific, here is an example.
Note that $(\mathbb{Z},+)$ is a group.
Definition 1. (Lang)
Let $G$ be a group and $S\subset G$.
If $G$ is the intersection of all subgroups $H$ of $G$ containing $S$, then "$\forall x\in S, x$ is called a generator of $G$.
And we say, $G$ is finitely generated iff there exists such $S$ which is finite
.
Definition2. (Rotman)
Let $(S|R)$ be a presentation for a group $G$. (This means that $G$ is isomorphic to a quotient of the free group $F(S)$ by the normal closure of $R$.
In this case $\forall x\in S$, $x$ is called a generator of $G$.
If $G$ has a presentation suh that $S$ is finite, then we call $G$ is finitely generated.
Which one is the usual definition?
Lang's definition is more usual than Rotman's..
More than that, usual equivalent condition we mean when we say finitely generated is
$G=\langle g_1,g_2,\ldots,g_n\rangle=\{g_1^{r_1}g_2^{r_2}\cdots g_n^{r_n} : r_i\in \mathbb{Z}\}$