It is straightforward to check that the discrete cube $Q=\{0,1\}^n$ with $\ell^1$-metric is 3-point-homogeneous, but not 4-point-homogeneous (assuming $n$ is large).
In other words, if $A\subset Q$ has at most 3 points, then any distance-preserving map $A\to Q$ can be extended to an isometry $Q\to Q$, but there is a 4-point set for which it does not hold.

I would like to see an example of $m$-point-homogeneous that is not $(m+1)$-point-homogeneous for $m\geqslant 4$.