Why is the texture mapped differently in these Two Cuboids? The left is default cube with one half removed using a Boolean Modifier. The right one I got using a second default cube and moving the bottom 4 vertices up a meter/unit. Both seem to have the "same" origin. Still, when using ProWood, the two give different results.

What am I getting wrong here?
In your image it looks like the texture mapping is created on the mesh prior to changes made by modifiers (whether it is generate or UV or whatever doesn't matter here). This is causing the texture to appear to be split in half on the one with the boolean.

If you clicked apply on the boolean modifier they should look the same.