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We present new neutral atomic carbon [CI](3P1-3P0) mapping observations within the inner ~7 kpc and ~4 kpc of the disks of NGC3627 and NGC4321 at a spatial resolution of 190 pc and 270 pc, respectively, using the ALMA Atacama Compact Array (ACA). We combine these with the CO(2-1) data from PHANGS-ALMA, and literature [CI] and CO data for two other starburst and/or active galactic nucleus (AGN) galaxies (NGC1808, NGC7469), to study: a) the spatial distributions of CI and CO emission; b) the observed line ratio RCICO = I_[CI](1-0)/I_CO(2-1) as a function of various galactic properties; and c) the abundance ratio of [CI/CO]. We find excellent spatial correspondence between CI and CO emission and nearly uniform RCICO ~0.1 across the majority of the star-forming disks of NGC3627 and NGC4321. However, RCICO strongly varies from ~0.05 at the centre of NGC4321 to >0.2-0.5 in NGC1808's starburst centre and NGC7469's centre with an X-ray AGN. Meanwhile, RCICO does not obviously vary with $U$, similar to the prediction of PDR models. We also find a mildly decreasing RCICO with an increasing metallicity over 0.7-0.85 solar metallicity, consistent with the literature. Assuming various typical ISM conditions representing GMCs, active star-forming regions and strong starbursting environments, we calculate the LTE radiative transfer and estimate the [CI/CO] abundance ratio to be ~0.1 across the disks of NGC3627 and NGC4321, similar to previous large-scale findings in Galactic studies. However, this abundance ratio likely has a substantial increase to ~1 and >1-5 in NGC1808's starburst and NGC7469's strong AGN environments, respectively, in line with the expectations for cosmic-ray dominated region (CRDR) and X-ray dominated region (XDR) chemistry. Finally, we do not find a robust evidence for a generally CO-dark, CI-bright gas in the disk areas we probed. (abbreviated)