As I found the solution, I share it to everyone.
I investigated some more with the web developer tool and found out that some 403 errors appeared when the browser tries to make POST to the lib/ajax directory within the moodle directory.
As I found it strange, a search on Google Gave me the explanation : such errors appear when an applicative Firewall on my web hosting provider was blocking the POSTs.
Desactivating the firewall made suddenly all activities additions possible !
Of course, I will keep this setting, because I want to make moodle work, but it is not recomanded because it decrease the overall security of the site (I have other sites running).
The Moddle developer Team could consider the following developer page on a very similar topic that explains how to systematically send the cookie (search for
{% csrf_token %} on the page). This would avoid to take the dangerous option of disabling the application firewall.Kind regards and Happy nw year.