Life Point costs for cards are not considered damage, so cards like "Wall of Revealing Light", "Final Countdown", and "Premature Burial" cannot have their costs re-directed by cards such as "Barrel Behind the Door" or negated by "Des Wombat".
A Direct attack is the process where a monster's whole ATK value is subtracted from the opponent's Life Points, either when there are no monsters on your opponent's side of the field, or if the attacking monster has the ability to attack directly, either by its own effect or the effect of another card. Damage from attacking directly is always treated as a Battle Damage.
Battle Damage is the damage inflicted to a player's Life Points from battle. Monster cards can still be destroyed even if Battle Damage is not inflicted. You do not usually take any Battle Damage from defense position Monsters, with the exception of piercing battle damage, where the difference between the attacking Monster's attack points and the defending Monster's defends points is subtracted from the opponents Life Points
Effect Damage is any damage that is not caused by battle (with the exception of "Gravekeeper's Vassal") or is a reduction.