Basically the simplified exhaust system is as follows:
Engine -> Header -> Catalytic Converter -> Downpipe -> Muffler
Axleback just means it's basically just the muffler.
The STI Sport Muffler is designed to connect to the stock midpipe, where the connector is at the left side.
Here's a picture of the stock catback vs the Fujitsubo Authorize R catback system, see how the midpipe connections are different (Fujitsubo has a Y joint design approaching the muffler):
Me personally I would want a system with the least amount of joints as possible, less points of failure causing a leak. JDM OEM optional exhaust has to meet JARI standards just like the stock car (gov emissions and noise requirements). Quality aftermarket JDM exhausts are usually JASMA certified quality standard and it's probably not as stringent in terms of noise and emissions (then again I thought JASMA was created to get gov authorization to stay legit), but then you should get more gains in performance.
You can research these orgs to get a better understanding:
JARI - Japan Automobile Research Institute (the place with the high speed loop track you always see on Best Motoring)
JASMA - Japanese Automotive Sports Muffler Association (jointly created by a bunch of aftermarket manufacturers)