No, the interplanar distance is d. It can't be 1/d because d has the dimensions of length in nλ=2dsinθ.
The distance d, as you say, is not the distance between atoms, but the separation between symmetry-equivalent planes of atoms, of which there are many sets.
In reciprocal space, the difference between the wavevectors of the incident and diffracted radiation is a reciprocal lattice vector whose magnitude is proportional to 1/d. So the smaller is d, the greater the angle of diffraction. The reciprocal lattice is an alternative representation of the information contained in the crystal structure, where each point represents a set of planes in the crystal, and therefore a spot in the XRD pattern.