Nope. What’s happened is that you have entered the correct password. Probably it’s a typo, and when you repeat the same typo it works.
The second password dialog is displayed when the sign in password does not work.The “You are already signed in with this password” message is shown when you re-enter the already tried and failed sign in password when prompted for a different password.
If you now got past that dialog, it means you typed something different from the sign in password, and that it’s also the correct password.
Our servers are not involved at all at this stage.