Hi, welcome! That looks like a good first project.
Hopefully you're already aware of issues with imperfect 90° inside corners, off level floors, etc. These are things that come up when installing cabinetry.
The good thing is that this is a closet, so all the imperfections are in a tiny room that doesn't get a lot of attention.
The first design keeps you from seeing the edge of the plywood top on the outside of the cabinet.
The dowelled construction isn't the best to prevent sagging, and books are heavy, but at only 20" wide, you probably will not have issues.
There are benefits of adding the back. It will hold the whole thing together so you can screw it to the back wall as well. You can nail through the back into the fixed shelves to help with sagging. Since the back instead of the shelves are holding the side panels together, you can add some adjustable shelves.
If you want it to look like part of the house, use matching baseboards to cover the bottom. This will also help hide the shims you will need under the left vertical panel to level the whole thing.
You can buy either 3/4" wide mouldings instead of edge banding to cover the edges of the plywood. If you're painting, you can glue and nail them on. If you're staining, glue them in place and hold them to the edge with strips of blue painter's tape.