After registering on your site and then signing out, here are my screenshots.
Reloading the page after logging out just continues to result in the same messages while I can still see my avatar there in the header.
After several minutes (after logging out) I reloaded the page again and this time I seem to be logged out.
So then I tried logging in again, and after that I still see the Login button…
Several minutes later I reload the page again but no change I am still not logged in.
I switch to Desktop mode (I am doing all this in Chrome on iPhone 13 Pro) and here is the screenshot…
Now several minutes later I tried again to Login but cannot login and continue to see the saw as the last screenshot.