As I said in the comment, it seems to me that your two definitions are not equivalent. For example, the first definition yields a convex set, while the second one does not. I sort of hope and suspect that it is the second one that you want, because it is the more interesting space.

I believe that your second definition describes a space homeomorphic to the geometric realisation of the poset of partitions of $\Lambda$. Let $M$ be a symmetric matrix satisfying your conditions 1-3. Suppose $0\le s \le 1$. Define a relation on $\Lambda$ by saying that $x\sim y$ if $M(x, y)\ge s$. Your conditions guarantee that it is an equivalence relation on $\Lambda$, i.e., a partition. Moreover, if $0\le s_1\le s_2\le 2$, then the partition associated to $s_2$ is a refinement of the partition associated to $s_1$. It follows that every matrix $M$ satisfying your conditions can be written uniquely as a convex combination of basic matrices associated to a nested chain of partitions of $\Lambda$. This is precisely saying that $\widetilde K$ is homeomorphic to the realization of the poset of partitions of $\Lambda$.

As I said in the comment, it seems to me that your two definitions are not equivalent. For example, the first definition yields a convex set, while the second one does not. I sort of hope and suspect that it is the second one that you want, because it is the more interesting space.

I believe that your second definition describes a space homeomorphic to the geometric realisation of the poset of partitions of $\Lambda$. Let $M$ be a symmetric matrix satisfying your conditions 1-3. Suppose $0\le s \le 1$. Define a relation on $\Lambda$ by saying that $x\sim y$ if $M(x, y)\ge s$. Your conditions guarantee that it is an equivalence relation on $\Lambda$, i.e., a partition. Moreover, if $0\le s_1\le s_2\le 2$, then the partition associated to $s_2$ is a refinement of the partition associated to $s_1$. It follows that every matrix $M$ satisfying your conditions can be written uniquely as a convex combination of basic matrices associated to a nested chain of partitions of $\Lambda$. This is precisely saying that $\widetilde K$ is homeomorphic to the realisation of the poset of partitions of $\Lambda$.

This is a contractible space, but is not a polyhedron. If it was a polyhedron, then its relative boundary would be homeomorphic to a sphere. As it stands, the boundary is homotopy equivalent to a wedge of sphere.

A couple of comments: First, a similar (but I think not homeomorphic) model for the partition complex was considered by Brenda Johnson in her paper "Derivatives of homotopy theory". This is the first paper where you can see a connection between the partition complex and Goodwillie calculus (but the partition complex is not named explicitly).

Second, this space is also closely related to the space of phylogenetic trees, which was studied for example by Biller, Holmes and Vogtmann in an influential paper called "Geometry of the space of phylogenetic trees".