Ok, finally it works (after 3 days and 20 hours of research)
I found a solution in the following post:
https://stackoverflow.com/questions/23842235/wordpress-jsonapi-wp-json-was-not-found-on-this-server
All I had to do (in my WordPress 6.1.1) was to go to “Settings / Permalinks” in the dashboard, and switch the Permalink structure to “post name”
