Which are the "9 ways to transpose"? In general there are 8 symmetries of a matrix, but you might also count reordering of rows/columns or blocks?

I agree that 1,5,9,4,8,3,7,2,6 pattern rotations is there somewhere, either in the columns or the rows of a solution, but that don't make all solutions the same.

Different people could reasonably disagree. The solutions with top rows:

159483726

594837261

Look pretty equivalent to me, which is how you get the 9 transpositions.

Yes, but the solutions are different if the order (permutation) of the rows with the pattern rotations are different. There is a huge number of ways the rows can be ordered (9!).