Regardless of the article....a quad tree when used in computer graphics or spatial partioniing is all about squares. A quad tree for spatial partioning will partition a region into 4 equal parts and each of those into 4 equal parts. The key is equal parts. An octree partitions a region into 8 equal parts. Quadtrees are often used in terrain based games and simulators like FSX. Octrees are often used in games for collision detection, lighting, etc. Sparse octrees can be used to represent an enormous amount of detail in voxel rendering. Quad trees are plane based and Octrees are usually volume based.
Reread the quad tree article, the regions don't have to be squares, though they often are, for simplicity in examples or teaching. Using rectangles adds a bit of complexity, but a very small one, IMO.