Rank identifications in join-meet ideals of thin distributive lattices

Let $L$ be a thin distributive lattice of rank $m+1$, and let $I_L$ be its join-meet ideal in the polynomial ring $S_L$. For $A\subseteq\{1,\dots,m\}$, let $J_A$ be the ideal obtained from $I_L$ by identifying, for every $i\in A$, the variables of the two elements of $L$ of rank $i$. We prove that $J_A$ is prime if and only if $A$ contains no turning rank and does not contain both ranks adjacent to a turning rank. In this case, $S_L/J_A$ is the toric ring obtained from Hibi's parametrization of the Hibi ring of $L$ by identifying suitable pairs of variables, and for an arbitrary $A$ we describe the saturation of $J_A$ as a lattice ideal.