in between I have solved my problem...or maybe the problem has just faded away form itself
The only change I did was:
- upgrading from 3.7 to 3.8 version
- (probably the solution?) setting up secure connexion, by switching to https for my hosting site (my hosting site was previously http, so I reset https following Moodle tutorial).
Now this works! It this works for you, I'd be happy to hear it!