For Facebook they do. They have a tight partnership. If Spotify did something wrong with their Facebook app, Facebook would have removed their app a long time ago.
Spotify started requiring a Facebook login from all new users on 22nd September 2011. On 6th December 2012 they put the non-FB signup option back. You signed up before they required FB so you never saw the requirement.
That is correct. It says quite clearly "Facebook Email or Spotify Username". It's the user's mistake for using their Facebook account instead of the account they just created.
Facebook and Spotify are tightly partnered together - at one point they actually REQUIRED the use of Facebook to log in. The option to register without Facebook was only reintroduced recently.
It says quite clearly: "Log in with Facebook or Spotify". That means you can log in using a Facebook account or a Spotify account.
The username field says "Facebook Email or Spotify Username". So when you type an email, you log in using a Facebook account.
It's not that hard to understand. By the way, that account you made on the sign up page is still unused: you logged in using a Facebook account, which is a different account from the one you just registered, so you have two spotify users now - one you signed up w/o Facebook and one you actually logged into.