Hello Simon, thanks for the answer.
So it seems really dumb, but as expected the Publishing Username wasn’t correct (I missed a space on the name). I fixed this and now I have the “You are connected to Discourse!” message.
Also no more error message on the SSO Provider tab. By the way I’m using WordPress 4.9.3.
Anyway, sorry for the inconvenience and thanks again for the answer. I’ll update this thread/create another one if I have any other issue especially with synchronizing the creation of posts between Wordpress and the forum.