We use new and updated gas and dust-corrected SFR surface densities to revisit the integrated star formation law for local "quiescent" spiral, dwarf, and low-surface-brightness galaxies. Using UV-based SFRs with individual IR-based dust corrections, we find that "normal" spiral galaxies alone define a tight $\Sigma (HI+H2)$-$\Sigma SFR$ relation described by a $n=1.41-0.07+0.07$ power law with a dispersion of $0.28-0.02+0.02$ (errors reflect fitting and statistical uncertainties). The SFR surface densities are only weakly correlated with HI surface densities alone, but exhibit a stronger and roughly linear correlation with H$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ surface densities, similar to what is seen in spatially-resolved measurements of disks. However, many dwarf galaxies lie below the star formation law defined by spirals, suggesting a low-density threshold in the integrated star formation law. We consider alternative scaling laws that better describe both spirals and dwarfs. Our improved measurement precision also allows us to determine that much of the scatter in the star formation law is intrinsic, and we search for correlations between this intrinsic scatter and secondary physical parameters. We find that dwarf galaxies exhibit second-order correlations with total gas fraction, stellar mass surface density, and dynamical time that may explain much of the scatter in the star formation law. Finally, we discuss various systematic uncertainties that should be kept in mind when interpreting any study of the star formation law, particularly the $X(CO)$ conversion factor and the diameter chosen to define the star-forming disk in a galaxy.