Well, I tried with cookies blocked and got the same behaviour, i.e. no log in and no message, while the app was still correctly added to my FB apps.
This time however, with the cookies unblocked, it all went smoothly, so this might well be the culprit. On top of that, having retried on my site with cookies unblocked as well, is also went well this time.
If I may and should this be confirmed, perhaps a test on blocked cookies in browsers triggering some message might help a bit.
Anyhow, thanks for your amazing reactivity and support.
Cheers, Bruno