Well, don't use the Peasant/Peon/Ghoul/Mur'gul Slave model

The lumber obtained shows up, because those 4 models have a stand animation with lumber attached on their hands. If you use any model but that, you'll get what you want.
To make harvesting faster, go to the tree you want in the Object Editor, Destructibles tab and scroll to Stats - Hit Points, set it to 5.00 for example or go to Abilities tab, search for "Harvest" and Data - Damage to Tree -> Increase damage dealt. So, if the tree has 5.00 hit points and the damage dealt is 5, the tree will fall in 1 attack.