It is not possible for the add-on to know whether or not you are successfully signed in to a particular site. (This is not a limitation of Password Depot, but is basically technically impossible for an add-on to know.)
This can lead to unwanted effects, in that even though you are already logged in, the add-on tries to fill in more form fields within the website.
You can solve this problem by limiting the login URL to exactly the URL where the login process takes place.
To do this, delete the template used, which could look as follows:
Then replace the URL with the exact value as shown here: