Three different logins screens???

Login simply takes you to wp-login

wp-login or wp-login.php uses a form which you can customise

http://codex.wordpress.org/Customizing_the_Login_Form

WP-login will always take you to this login page

WP-admin will take you to the login page if you’re not logged in, but straight to your site admin if you have ticked the “remember me” box, without showing you the login form.

So each serves a purpose. Login is easy to remember, but actually uses wp-login.php as the file, which you can call directly and/or customise, and wp-admin lets you skip logging in if you’ve clicked the remember me box.

Hope that helps