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@M.Winter My apologies for being somewhat imprecise. I have now included a screenshot of the relevant 2 half pages of the book. In particular, you can see how both conditions come into play: commutativity corresponds to the permutation character being multiplicity-free, whereas symmetry corresponds to the 2-orbits being self-paired.
I assume that the difference lies in your definition of association scheme: Bannai and Ito consider not necessarily commutative association schemes, whereas other sources assume commutativity as part of the definition. Does this help?
Not as easy as the above example, but the one-page paper "On Isomorphisms between Coxeter Groups" by Bernhard Mühlherr (doi.org/10.1023/A:1008347930052) gives an explicit example of two non-isomorphic irreducible Coxeter systems of rank 4 for which the resulting groups are isomorphic.
@WolfgangTintemann: Notice that the formulas on the Wikipedia page are only partially defined: $0 \cdot \infty$ and $\infty + \infty$ are undefined. So this does not make $\mathbf{P}^1_K$ into a ring.
I'm not sure I understand your last comment. If you have a solution with $a_i \in \{ -1, 1\}$, then you also have a solution with $a_i \in \{ 0, 1\}$ simply by replacing each $a_i$ with $(a_i + 1)/2$ (using the fact that setting all $a_i = 1$ is also a valid solution). Am I misunderstanding something?
Notice that this description does not allow to recover the original object from the resulting directed graph alone: you have to "remember" the description of the elements of $V$.
