Definition 1:
A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$
Definition 2:
A set $S \subset M$ is bounded if $\exists r > 0, \forall x,y \in S, d(x,y) \leq r$
It seems the first definition is saying that the set $S$ can be put into some ball. The second definition is saying that elements within the set are close to each other. So intuitively they seem to be different.
Are the two definitions equivalent? If not, what is the standard definition for a set being bounded?
They are equivalent. Suppose your first statement holds, then $\forall y_1,y_2 \in S$, $$ d(y_1,y_2) \leq d(x,y_1)+d(x,y_2) < 2r. $$ Conversely, suppose your second statement holds, then $\forall x \in S$, $S\subset B_x(2r)$.
Your second definition is more commonly seen in literature. There is, however, a different definition of boundedness in general topological vector spaces. That is:
The subset $E$ of a topological vector space $X$ is bounded if to every neighbourhood $V$ of $0$ in $X$ corresponds a number $s>0$ such that $E\subset tV$ for all $t>s$.
This topology-boundedness can differ from metric-boundedness, even if the topology is induced by the metric.