The diagonals of a quadrilateral bisect each other at right angles, but the two diagonals have different lengths. What is the most precise name for this quadrilateral?

Diagonals that bisect each other make the figure a parallelogram; adding that they cross at right angles makes all four sides equal, so it is a rhombus. It is not a square, because a square has equal diagonals and these differ. A rectangle also has equal diagonals, so that is out, and in a kite that is not a parallelogram only ONE diagonal bisects the other.