Definition of amenable group

Yes, this is correct.

As Yves Cornulier notes, your condition implies the Følner condition which is the same as yours, but with the condition $A \subseteq \langle X \rangle$ removed.

Conversely, suppose $G$ is amenable. Then the subgroup $H = \langle X \rangle$ is amenable and if $(F_n)_{n \in \mathbb{N}}$ is a Følner sequence for $H$ then choosing $n$ large enough allows us to take $A = F_n$.

An advantage of your condition is that it is immediately clear that a subgroup of a group satisfying your condition also satisfies your condition. To show that the usual Følner condition passes to subgroups requires an additional argument.