We re-examine the classifications of supernovae (SNe) presented in the Lick Observatory Supernova Search (LOSS) volume-limited sample with a focus on the stripped-envelope SNe. The LOSS volume-limited sample, presented by Leaman et al. and Li et al., was calibrated to provide meaningful measurements of SN rates in the local universe; the results presented therein continue to be used for comparisons to theoretical and modeling efforts. Many of the objects from the LOSS sample were originally classified based upon only a small subset of the data now available, however, and recent studies have both updated some subtype distinctions and improved our ability to perform robust classifications, especially for stripped-envelope SNe. We re-examine the spectroscopic classifications of all events in the LOSS volume-limited sample (180 SNe and SN impostors) and update them if necessary. We discuss the populations of rare objects in our sample including broad-lined SNe Ic, Ca-rich SNe, SN 1987A-like events (we identify SN 2005io as SN 1987A-like here for the first time), and peculiar subtypes. The relative fractions of SNe Ia, SNe II, and stripped-envelope SNe in the local universe are not affected, but those of some subtypes are. Most significantly, after discussing the often unclear boundary between SNe Ib and Ic when only noisy spectra are available, we find a higher SN Ib fraction and a lower SN Ic fraction than calculated by Li et al.: spectroscopically normal SNe Ib occur in the local universe 1.7 ± 0.9 times more often than do normal SNe Ic.