What does "if and only if" mean in definitions? Consider the following definition:

A sequence $\{p_n\}$ is Cauchy if we have that for every $n, m \ge N$: $$|p_n - p_m| < \epsilon$$

Although if and only if is not used, we know that if a sequence is Cauchy, the Cauchy criterion holds. Consider now the following definition:

An action of a group $G$ on a set $X$ is transitive if and only if for each $x, y \in X$ there is a $g \in G$ such that $gx = y$.

What motivates the use of "if and only if" here? As far as definitions go, my head has automatically thought "if and only if." The latter example is taken from Ratcliffe's "Foundations of Hyperbolic Manifolds", and he switches a lot between "if" and "if and only if" in his definitions.

There is no difference between "if" and "if and only if" in a definition. More precisely, in the context of a definition "if" should be read as a shorthand for "if and only if".
I can't comment about Ratcliffe specifically, but my guess is that the reason it appears E is using them interchangeably is because E is.
