Since the discovery of cuprate superconductors, the role of the apical oxygen has been debated, and a putative correlation between the Cu-apical-O distance and the critical temperature was proposed. Recent scanning tunneling microscopy measurements (O’Mahony et al., PNAS 119 (2022)) exploiting the natural superstructure modulation of the cuprate superconductor Bi₂Sr₂CaCu₂O_8+δ (Bi-2212) have revealed a possible correlation between the apical distance and the density of Cooper pairs, thus reviving this debate. Leveraging a combination of density-functional theory and cluster dynamical mean-field theory, we compute from first principles the variations of pair density induced solely by apical oxygen displacement in Bi₂Sr₂CuO_6+δ, Bi-2212, and HgBa₂CuO_4+δ. Our approach reproduces quantitatively with the measured variations of pair density. It enables a microscopic understanding based on variations of effective hole-doping in the CuO₂ planes. Moreover, it provides an absolute scale to the pair-density modulations by apical oxygen displacement, thus revealing that they are modest in comparison to other structural changes. This warrants caution in interpreting correlations between T_c and apical distance inferred from comparisons across different cuprate compounds. In summary, this work demonstrates the capability of ab initio approaches to provide quantitative and accurate guidelines to stimulate the search for improved high-temperature superconductors.