Resonant inelastic X-ray scattering investigation of Hund's and spin-orbit coupling in $5d^2$ double perovskites

B site ordered $5d^2$ double perovskites ($\mathrm{A_2BB'O_6,\ B'}=5d^2)$ display a remarkable range of physical properties upon variation of the chosen B and $\mathrm{B'}$ site ions. This sensitivity to chemical substitution reflects the delicate balance and profound impact of strong electronic correlation and spin-orbit coupling in such systems. We present rhenium $L_2$ and $L_3$ resonant inelastic X-ray scattering (RIXS) measurements of two such physically dissimilar materials, Mott-insulating $\mathrm{Ba_2YReO_6}$ and semiconducting $\mathrm{Sr_2CrReO_6}$. Despite these differences, our RIXS results reveal similar energy scales of Hund's ($J_H$) and spin-orbit coupling ($ζ$) in the two materials, with both systems firmly in the intermediate Hund's coupling regime where $J_H/ζ\sim 1$. However, there are clear differences in their RIXS spectra. The conductive character of $\mathrm{Sr_2CrReO_6}$ broadens and obfuscates the atomic transitions within an electron-hole continuum, while the insulating character of $\mathrm{Ba_2YReO_6}$ results in sharp atomic excitations. This contrast in their RIXS spectra despite their similar energy scales reflects a difference in the itinerancy-promoting hopping integral and illustrates the impact of the local crystal environment in double perovskites. Finally, $L_2$ and $L_3$ edge analyses of the atomic excitations in $\mathrm{Ba_2YReO_6}$ reveal that the ordering of the low lying excited states is inverted compared to previous reports, such that the appropriate energy scales of Hund's and spin-orbit coupling are significantly modified. We present exact diagonalization calculations of the RIXS spectra at both edges which show good agreement with our results for new energy scales of $ζ=0.290(5)$ eV and $J_H=0.38(2)$ eV ($J_H/ζ=1.30(5)$).