Equivalence of definitions of Krull dimension of a module

These definitions are not the same in general, if $M$ is not f.g.

Consider the module $\mathbb Q_p/\mathbb Z_p$ over $\mathbb Z_p$. Its annihilator is $0$, so the first definition gives dimension $1$. On the other hand, its support is the closed point of Spec $\mathbb Z_p$, and so the second definition gives dimension $0$.

If you are reading an article that applies the notion of dimension in the non-f.g. context, then you will either have to look and see if the author defines their terms, or else determine from the context (e.g. how they argue) which definition is in use.