I've heard some fairly nasty things about hail from LP storms as well...a fairly innocent-looking thin precip core can contain damaging hail. And don't count out HP tornadoes...Plainfield IL 1990 (F5) was so shrouded in precip as to be invisible as a funnel, and Nashville 1998 (F3) looked about twice as wide as it really was. Many of the WI storms on August 18, 2005 cycled back and forth from HP to classic.
I would agree with the definition that a "classic" supercell has a distinct separation between precip area and updraft area, as well as the dominant updraft/meso area at the back right flank (instead of forward flank as in many HPs), but still maintaining a stout, visible precip core (unlike LPs). These storms can last the longest and produce the most severe weather because of this balance between updraft and downdraft, inflow and outflow.