Inspired by our serendipitous discovery of six active galactic nuclei (AGNs) with varying broad-H alpha fluxes over years from our search for intermediate-mass black holes (IMBHs), we conduct a systematic investigation of changing-look (CL) and large-variability AGNs. We collect all the CL AGNs at z alpha and perform careful decomposition fittings to both their images and spectra. We find two observational facts: (1) The host galaxies of local CL and large-variability AGNs, mainly being Seyferts, are in the red (gas-poor) tail of the general Seyfert galaxy population. (2) In contrast, there is a significant trend that their more luminous counterparts, namely CL and extremely variable quasars (CLQs and EVQs), are different: CLQs are generally in blue galaxies; in terms of the diagram of star formation rate and M (*), local CL Seyfert galaxies are located in the green valley, whereas CLQ hosts are in the star-forming main sequence. We propose explanations for those strongly variable Seyferts and quasars, respectively, under the assumption that accretion disks broadly depend on nuclear fueling modes. Local large-variability and CL Seyferts are in nuclear famine mode, where cold-gas clumps can be formed stochastically in the fueling flow, and their episodic infall produces sharp peaks in the accretion-rate curve. CLQs and EVQs are in feast fueling mode, which may account for both their preference for blue galaxies and their variability pattern (high-amplitude tail of the continuous distribution). Lastly, we propose a new thinking: to search for IMBHs by optical variability in red galaxies.