Revolving Door has many parallel storylines. You can read it along any route, or jump between routes (the canon order does this), as long as every chapter in a certain route is read in sequential order. A play button denotes the start chapter of a route (a possible entry point for reading); a record button denotes the most recent chapter of each route. Both icons together denotes a currently standalone chapter. If you prefer the original story map, you can find that here.