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 $\Lambda$.