We study the distribution of cold molecular gas in the circumunuclear disks (CND; r ≤ 200 pc) of a sample of 64 nearby (DL = 7 ‑ 45 Mpc) disk galaxies – including 45 active galactic nuclei (AGN) and 19 nonAGN – for which high-spatial-resolution (median value ≃36 pc) multiline CO interferometer observations have been obtained at millimeter wavelengths with the Atacama Large Millimeter Array (ALMA) and/or Plateau de Bure Interferometer (PdBI). We decipher whether or not the concentration and normalized radial distribution of cold molecular gas change as a function of X-ray luminosity in the 2–10 keV range (LX) in order to analyze the imprint left by AGN feedback. We also look for similar trends in the concentration and normalized radial distribution of the hot molecular gas and in the hot-to-cold-molecular gas mass ratio in a subset of 35 galaxies using near-infrared (NIR) integral field spectroscopy data obtained for the H2 1-0 S(1) line. We find a significant turnover in the distribution of the cold molecular gas concentration as a function of X-ray luminosity with a breakpoint that divides the sample into two branches: (1) the "AGN build-up branch" (LX ≤ 1041.5 ± 0.3 erg s‑1) and (2) the "AGN feedback branch" (LX ≥ 1041.5 ± 0.3 erg s‑1). Lower-luminosity AGN and nonAGN of the AGN build-up branch show high cold molecular gas concentrations and centrally peaked radial profiles on nuclear (r ≤ 50 pc) scales. Higher-luminosity AGN of the AGN feedback branch show a sharp decrease in the concentration of molecular gas and flat or inverted radial profiles. The cold molecular gas concentration index (CCI) – defined as the ratio of surface densities at r ≤ 50 pc (Σ50gaz) and r ≤ 200 pc Σ200gaz), namely CCI ≡ log10(Σ50gaz/Σ200gaz)) – spans a 0.63 dex range, equivalent to a factor ≃4–5, between the galaxies lying at the high end of the AGN build-up branch and the galaxies showing the most extreme nuclear-scale molecular gas deficits in the AGN feedback branch. The concentration and radial distributions of the hot molecular gas in our sample follow qualitatively similar but less extreme trends as a function of X-ray luminosity. As a result, we find higher values of hot-to-cold molecular gas mass ratios on nuclear scales in the highest luminosity AGN sources of the AGN feedback branch. These observations confirm – with a three times larger sample – previous evidence found in the context of the Galaxy Activity Torus and Outflow Survey (GATOS) that the imprint of AGN feedback on the CND-scale distribution of molecular gas is more extreme in higher luminosity Seyfert galaxies of the local Universe.