We present resolved [CI] line intensities of 18 nearby galaxies observed with the SPIRE FTS spectrometer on the Herschel Space Observatory. We use these data along with resolved CO line intensities from $J_mathrm{up} = 1$ to 7 to interpret what phase of the interstellar medium the [CI] lines trace within typical local galaxies. A tight, linear relation is found between the intensities of the CO(4-3) and [CI](2-1) lines; we hypothesize this is due to the similar upper level temperature of these two lines. We modeled the [CI] and CO line emission using large velocity gradient models combined with an empirical template. According to this modeling, the [CI](1-0) line is clearly dominated by the low-excitation component. We determine [CI] to molecular mass conversion factors for both the [CI](1-0) and [CI](2-1) lines, with mean values of $alpha_{mathrm{[CI](1-0)}} = 7.3$ M$_{mathrm{sun}}$ K$^{-1}$ km$^{-1}$ s pc$^{-2}$ and $alpha_{mathrm{[CI](2-1)}} = 34 $ M$_{mathrm{sun}}$ K$^{-1}$ km$^{-1}$ s pc$^{-2}$ with logarithmic root-mean-square spreads of 0.20 and 0.32 dex, respectively. The similar spread of $alpha_{mathrm{[CI](1-0)}}$ to $alpha_{mathrm{CO}}$ (derived using the CO(2-1) line) suggests that [CI](1-0) may be just as good a tracer of cold molecular gas as CO(2-1) in galaxies of this type. On the other hand, the wider spread of $alpha_{mathrm{[CI](2-1)}}$ and the tight relation found between [CI](2-1) and CO(4-3) suggest that much of the [CI](2-1) emission may originate in warmer molecular gas.
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