We reformulate the overlap of wave-function derivatives that defines the atomic axial tensors (AATs) and velocity-gauge atomic polar tensors (VG-APTs) used to compute vibrational circular dichroism (VCD) rotatory strengths. In place of our earlier determinant-level derivatives, the new approach expresses these tensors as contractions involving molecular-orbital derivative integrals, perturbed and unperturbed one- and two-electron density matrices, and dot products of the perturbed correlated wave-function amplitudes. Following the elegant treatment of the diagonal Born-Oppenheimer correction by Gauss and co-workers, the formulation holds for any real, normalized CI-like wave function. The two-electron density contribution vanishes exactly, because one perturbation is real (nuclear) and the other imaginary (magnetic field or momentum); this removes multiple terms from our earlier AAT and VG-APT expressions for second-order Møller-Plesset perturbation theory (MP2) and configuration interaction singles and doubles (CISD). We also present a simplified form of the origin-independent length-gauge [LG(OI)] rotatory strength that avoids the singular-value decomposition of the outer product of the length-gauge APT and VG-APT vectors for each mode, and that clarifies how the LG(OI) and velocity-gauge rotatory strengths are related. Applying the density-based AATs and VG-APTs to (P)-hydrogen peroxide, (S)-methyloxirane, and (S)-3-chloro-1-butyne, we find that discrepancies among the gauges arise chiefly from basis-set incompleteness in the VG-APT. The charge-sum rule serves as a useful measure of that incompleteness, independent of the correlation method.