Extended extragalactic radio sources have traditionally been classified into Fanaroff & Riley (FR) I and II types, based on the ratio r s of the separation S between the brightest regions on either sides of the host galaxy and the total size T of the radio source (r s ≡ S/T ). In this paper, we examine the distribution of various physical properties as a function of r s of 1040 luminous (L L * ) extended radio galaxies (RGs) at z < 0.3 selected with well-defined criteria from the SDSS, NVSS, and FIRST surveys. About 2/3 of the RGs are lobe dominated (LD) and 1/3 have prominent jets. If we follow the original definition of the FR types, i.e., a division based solely on r s , FR I and FR II RGs overlap in their host galaxy properties. However, the rare LD sources with r s 0.8 and [O iii] λ5007 line luminosity > 10 6 L are markedly different on average from the rest of the RGs, in the sense that they are hosted in lower mass galaxies, live in relatively sparse environments, and likely have higher accretion rates onto the central supermassive black hole (SMBH). Thus, these high emission line luminosity, high-r s LD RGs, and the rest of RGs form a well-defined dichotomy. Motivated by the stark differences in the nuclear emission line properties of the RG subsamples, we suggest that the accretion rate onto the SMBH may play the primary role in creating the different morphologies. At relatively high accretion rates, the accretion system may produce powerful jets that create the "classical double" morphology (roughly corresponding to the LD sources with r s 0.8 and emission lines); at lower accretion rates, the jets from a radiatively inefficient accretion flow generate radio lobes without apparent "hot spots" at the edge (corresponding to the majority of LD sources). At slightly lower accretion rates and in galaxies with dense galactic structure, sources with prominent jets result. It is possible that while the high accretion rate systems could affect sub-Mpc scale environments, the jets from lower accretion rate systems may efficiently suppress activity within the host galaxies.