I did it a bit different. I went to account management and entered the code XXXX-XXXX-XXXX on the back of my booklet into the redeem codes section. This allowed me to download the client, but I already had it bacause of beta and early access so I canceled the download. Then I went and logged on through the FFXIV disc (not the old client) and got to the log in screen (where all the developer's messages are). I logged in there and it made me accept the contract agreement again. Then it made me register my game to my SE account. I did not add a new account through this process, but instead proceeded with my old account that was already selected.
So no, you do not have to first log onto the browser to register the game before the PS3 will let you register.