Let $X$ be a set. Write $M(X)$ for the set of all functions $d:X\times X\to [0,\infty]$ that endow $X$ with the structure of a generalized metric space (i.e., $d(x,x)=0 $ and the triangle inequality). Let $M_S(X)$ be the set of all functions $d:X\times X\to X$ that endow $X$ with the structure of a metric space (e.g., $d(x,y)=d(y,x)$ also holds).
Define an order on each of these sets by declaring $d\le e$ precisely if the identity $(X,d)\to (X,e)$ is a non-expanding function. It is very easy to see that with this ordering each of $M(X)$ and $M_S(X)$ is a complete lattice and that $M_S(X)$ is a sub complete meet semilattice of $M(X)$.
Do these lattices have a common name? Are their basic properties known? Ideally, is there a lattice theoretic characterization on a lattice $L$ that is equivalent to $L$ being isomorphic to a sublattice of one of those?