A group $\mathbb{G}$ is called rational (https://groupprops.subwiki.org/wiki/Rational_group) if $g,g' \in \mathbb{G}, \langle g \rangle = \langle g' \rangle \Rightarrow \exists x \in \mathbb{G}: xgx^{-1} = g'$.
I followed, that $\mathbb{Z}$ is not rational, because e.g. $\langle 2 \rangle = \langle -2 \rangle$, but $x+2+(-x)=2 \neq -2$.
On the other hand, I am reading a paper, saying 'a group X is called rational if it is isomorphic to a subgroup of the group of rationals $\mathbb{Q}$', and this would imply that $\mathbb{Z}$ is rational.
Are the defintions misleading or is something wrong with my conclusions?