A necessary and sufficient condition for a structure to be rigid.

Question

Is this a necessary and sufficient condition for a structure $M$ to be rigid: For all distinct $m, m'$ in the underlying set of $M$, $Th(M,m) \neq Th(M,m')$?

Answer 1

As Asaf points out in the comments, it is entirely straightforward that if $\text{Th}(M,m)\neq \text{Th}(M,m')$ for all distinct $m$ and $m'$ in $M$, then $M$ is rigid.

The converse is not true. Hint: Every ordinal $(\alpha,<)$ is rigid.