As much as I hate the overcomplicated login process, this is actually as straightforward as it can get.
  • Member Name: Enter anything here. This name will never be seen anyone else, it's just to distinguish between the locally saved accounts.
  • PlayOnline ID: Eight characters, four letters, four digits. It's the one that was randomly generated for you.
  • Set Password: If this is changed to "Save" (default is "Not set"), you will not need to enter the password every time you log in. However, to my knowledge this is not possible to not set this anymore when using an SE account anymore, which is mandatory, so I don't know why this option still exists. Maybe for trial accounts? Not sure, maybe someone else knows.
  • PlayOnline Password: You can only enter this if you changed the above to "Save". If not you'll be prompted to enter it when you log in.
  • Member Password: This is only used locally and has nothing at all to do with POL, FFXI or anything else. It's only to stop vile house mates to log in as you and draw cock pictures on your mog house walls. If you don't have anyone like that, I strongly recommend not using this, as it only leads to confusion. To unset a member password, you will need to go to the settings again and delete what's currently in there. Just save the settings with this blank. If you forgot your member password, you'll probably need to reinstall the POL client.
  • Square Enix ID: This is the name of your SE account, whatever you chose it to be. This is your umbrella account, one SE account can contain several POL accounts, and each POL account can contain several characters. This is what you log in to the SE Account Management site with.
  • One-Time Password: The password displayed on the security token. Obviously it also can't be set permanently but has to be entered every time it's asked of you when you log in. To disable it, just set it to "Do Not Use" (default, otherwise select "Use".

There is no way to save the SE password, you enter that every time you log in. It's the same password you use at SE the Account Management site. When you then click on "Log In", you will need to enter your SE password only (because the POL password had to be saved in the settings), and possibly the one-time password from the security token, if you use that.