SIEGE is pretty self explanatory, it's what siege weapons use
STRUCTURAL - yup, that's defensive towers.
CHOP - bonus damage against units without heavy armor upgrade, units like vanilla berserkers use this.
URUK - bonus damage against units with heavy armor upgrade, units like upgraded berserkers use this.
FLAME - fire damage like Saruman's fireball or fire arrows on upgraded archers.
FROST - ice damage on units like Carn Dum Archers with frost arrows.
WATER - monsters use this damage type, so Trolls, Werewolves...
LOGICAL_FIRE - afaik this is dealt when the ground is on fire, not sure on this one though. Not too relevant anymore because you can't set the ground on fire on most maps.
CAVALRY_RANGED - basically armor piercing arrows, good against units with heavy armor although not as strong as URUK.
CAVALRY is only the melee damage, you should maybe clarify that to avoid misunderstandings.
POISON doesn't do damage over time by definition afaik, that's a different, special property of those weapons. That's just nitpicking though^^
SLASH, PIERCE, CRUSH, CAVALRY, HERO, HERO_RANGED, MAGIC, SPECIALIST and the ones I listed are commonly used in Edain atm. The other ones (and logical fire, to an extent) are to my knowledge more or less irrelevant unless they are used on other custom maps^^
Hope this helped