A uniformity is separated if and only if its induced topology is Hausdorff

Suppose $\mathcal U$ is separated and $x\ne y$. Choose $E$ with $(x,y)\notin E$, then a symmetric $D$ with $D\circ D\subseteq E$.

Conversely, if the induced topology is Hausdorff and $x\ne y$, choose disjoint neighbourhoods of $x,y$ and refine the first by an entourage ball $E[x]$; then $y\notin E[x]$, so $(x,y)\notin E$.

The neighbourhoods $D[x]$ and $D[y]$ are disjoint: if $z$ belonged to both, symmetry would give $(x,z),(z,y)\in D$ and hence $(x,y)\in D\circ D\subseteq E$.

Thus the induced topology is Hausdorff by [L2].

Every distinct pair is excluded by an entourage, so $\mathcal U$ is separated by [A1].