The determinant is not just “a computing recipe to memorize.” In 2D, it is the area of the parallelogram spanned by two vectors, and in 3D it is the volume of the parallelepiped spanned by three vectors. So a determinant of 0 = flattened = linearly dependent. The rank counts how far the vectors have been flattened, and from there it leads, in one unbroken thread, to the shape of the solutions of simultaneous equations (point, line, plane).
Mochizuki Research Group, Tokyo University of Science / From the lecture notes “(16) What does a zero or nonzero determinant mean?” (in Japanese). The derivation is on the explanation page (in Japanese).