  Existing star-forming vs. active galactic nucleus (AGN) classification
schemes using optical emission-line diagnostics mostly fail for low-metallicity
and/or highly star-forming galaxies, missing AGN in typical $z\sim0$ dwarfs. To
recover AGN in dwarfs with strong emission lines (SELs), we present a
classification scheme optimizing the use of existing optical diagnostics. We
use SDSS emission-line catalogs overlapping the volume- and mass-limited
RESOLVE and ECO surveys to determine the AGN percentage in SEL dwarfs. Our
photoionization grids show that the [O III]/H$\beta$ versus [S II]/H$\alpha$
diagram (SII plot) and [O III]/H$\beta$ versus [O I]/H$\alpha$ diagram (OI
plot) are less metallicity sensitive and more successful in identifying dwarf
AGN than the popular [O III]/H$\beta$ versus [N II]/H$\alpha$ diagnostic (NII
plot or "BPT diagram"). We identify a new category of "star-forming AGN"
(SF-AGN) classified as star-forming by the NII plot but as AGN by the SII
and/or OI plots. Including SF-AGN, we find the $z\sim0$ AGN percentage in
dwarfs with SELs to be $\sim$3-16\%, far exceeding most previous optical
estimates ($\sim$1\%). The large range in our dwarf AGN percentage reflects
differences in spectral fitting methodologies between catalogs. The highly
complete nature of RESOLVE and ECO allows us to normalize strong emission-line
galaxy statistics to the full galaxy population, reducing the dwarf AGN
percentage to $\sim$0.6-3.0\%. The newly identified SF-AGN are mostly gas-rich
dwarfs with halo mass $ < 10^{11.5} M_\odot$, where highly efficient cosmic gas
accretion is expected. Almost all SF-AGN also have low metallicities (Z
$\lesssim 0.4$ Z$_\odot$), demonstrating the advantage of our method.
