The $<$-relation on $\mathbb{Z}$ is not definable in $(\mathbb{Z}, 0, +)$

The situation is invariant under negation. It follows that you can't distinguish $\lt$ from $\gt$.

It is easy to check that $h:\mathbb{Z}\to\mathbb{Z}$ via $x\mapsto -x$ is an automorphism and $\mathbb{Z}^+$, the set of positive integers, is not closed under this automorphism. Hence $\mathbb{Z}^+$ is not definable in $(\mathbb{Z},0,+)$. Note that the definability of $<$-relation is the same as $\mathbb{Z}^+$.