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We present new Chandra X-ray observations of seven low-mass black holes ( {M}_{BH}~{10}^{6} {M}_☉ ) accreting at low-bolometric Eddington ratios between -2.0\lesssim logL_{bol}/L{Edd}\lesssim -1.5 . We compare the X-ray properties of these seven low-mass active galactic nuclei (AGNs) to a total of 73 other low-mass AGNs in the literature with published Chandra observations (with Eddington ratios extending from -2.0\lesssim logL_{bol}/L{Edd}\lesssim -0.1 ). We do not find any statistical differences between the low and high Eddington ratio low-mass AGNs in the distributions of their X-ray to ultraviolet luminosity ratios ( α_{ox} ), or in their X-ray spectral shapes. Furthermore, the α_{ox} distribution of low- L_{bol}/L{Edd} AGNs displays an X-ray weak tail that is also observed within high- L_{bol}/L{Edd} objects. Our results indicate that between -2\lesssim logL_{bol}/L{Edd}\lesssim -0.1 , there is no systematic change in the structure of the accretion flow for active galaxies hosting {10}^{6} {M}_☉ black holes. We examine the accuracy of current bolometric luminosity estimates for our low- L_{bol}/L{Edd} objects with new Chandra observations, and it is plausible that their Eddington ratios could be underestimated by up to an order of magnitude. If so, then in analogy with weak emission line quasars, we suggest that accretion from a geometrically thick, radiatively inefficient "slim disk" could explain their diverse properties in α_{ox} . Alternatively, if current Eddington ratios are correct (or overestimated), then the X-ray weak tail would imply that there is diversity in disk/corona couplings among individual low-mass objects. Finally, we conclude by noting that the α_{ox} distribution for low-mass black holes may have favorable consequences for the epoch of cosmic reionization being driven by AGN.