"Pure mazes" are simply that, just mazes. Each type of pure maze may have some minor rules associated with it, but the conceptual layout of all the possible problem states (called the problem space in cognitive science) matches the physical layout of the maze (which I call the physical problem space). In other words, all of the possible places you can be in the maze matches exactly to the physical layout of the maze.
"Mazes with rules" are more complicated mazes, where the problem space no longer matches the physical problem space. You could be at the same physical spot in the maze, but you could be at different points in the puzzle, depending on how you entered, or if you had been there before, or the shape of the allowable move, etc.... The physical layout is still a very important aspect of these puzzles.
"Puzzles with layout" have even less connection between their problem space and their physical problem space. In these puzzles, the physical space you move around in is more open, and less like a maze. Here the rules which govern your movement around the space provide more of the framework for the problem, instead of physical barriers. Some of these puzzles are solved not by crossing a pre-defined finish line, but by satisfying a condition or set of conditions.
"Pure puzzles" have no discernable maze-like qualities, and are simply puzzles which rely on rules. Some use physical layouts to define the problem, but do not require you to navigate a physical space from start to finish.
A fifth category of "text puzzles" has now been added. While these will still require logical thinking, they are mostly word-based puzzles. The logic puzzles have been moved here and join the new boggling stories.