We analyze the rest-optical emission-line spectra of $zsim2.3$ star-forming galaxies in the complete MOSFIRE Deep Evolution Field (MOSDEF) survey. In investigating the origin of the well-known offset between the sequences of high-redshift and local galaxies in the [O III]5008/H$beta$ vs. [N II]6585/H$alpha$ ([N II] BPT) diagram, we define two populations of $zsim2.3$ MOSDEF galaxies. These include the high population that is offset towards higher [O III]5008/H$beta$ and/or [N II]6585/H$alpha$ with respect to the local SDSS sequence and the low population that overlaps the SDSS sequence. These two groups are also segregated within the [O III]5008/H$beta$ vs. [S II]6718,6733/H$alpha$ and the [O III]4960,5008/[O II]3727,3730 (O$_{32}$) vs. ([O III]4960,5008+[O II]3727,3730)/H$beta$ (R$_{23}$) diagram, which suggests qualitatively that star-forming regions in the more offset galaxies are characterized by harder ionizing spectra at fixed nebular oxygen abundance. We also investigate many galaxy properties of the split sample and find that the high sample is on average smaller in size and less massive, but has higher specific star-formation rate and star-formation-rate surface density values and is slightly younger compared to the low population. From Cloudy+BPASS photoionization models, we estimate that the high population has a lower stellar metallicity (i.e., harder ionizing spectrum) but slightly higher nebular metallicity and higher ionization parameter compared to the low population. While the high population is more $alpha$-enhanced (i.e., higher $alpha$/Fe) than the low population, both samples are significantly more $alpha$-enhanced compared to local star-forming galaxies with similar rest-optical line ratios. These differences must be accounted for in all high-redshift star-forming galaxies -- not only those offset from local excitation sequences.
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