I edited my previous post in a burst of enthusiasm that was unfortunately short-lived. No, I did not actually figured it out yet. Problem remains. Here it is:
I've installed Fancier Author Box on my French website and proceeded to translate it using Poedit. I created .po and mo. files and uploaded them in the languages folder. Sadly, nothing changed. I even deleted the original .mo and .pot files in English to see what happened: nothing. Despite the English language files being nonexistent, the plugin remains 100% in English, including when I access the website from a browser set in French (some plugins directly match your browser's language instead of the one set in wp-config.php).
I don't think I made a mistake when following the steps for the translation. WordPress and most of the other plugins on the website already work fine in their French version. However, you may find interesting that another plugin, which I've installed today, and contained many ready translations (including in French), also inexplicably remains in English. I am really at a loss here.
Has anyone ever reported such a problem? What can I do? I really like this plugin in any other way.