Is the additive group of integers a rational group?

Not all definitions are standard. Definitions may vary depending on the text book or author (for example: how many definitions of continuity we already have? they are not always equivalent). Rational group can be one of them (after all the term is not widely popular). Check the paper for the definition.

Side note: according to the first definition if an abelian group is rational then every element has to be of order $2$. That follows because if $g\in A$ is of order greater than $2$ then the Euler totient function tells us that there is some other power of $g$ that generates $\langle g\rangle$. And that element cannot be conjugate to $g$ because $A$ is abelian. In particular abelian rational groups are vector spaces over $\mathbb{Z}_2$ and therefore are of the form $\bigoplus \mathbb{Z}_2$ and indeed every such group is rational.

These must be two unrelated definitions. After all, since $\mathbb{Q}$ is abelian, every conjugation moves no elements.