psql could not connect to server after complete reinstallation [Ubuntu 21.10 / PostgreSQL 13]

Hi,

After experiencing issues related to the existence of two different 
PostgreSQL versions installed on the same machine, I decided to do a 
"remove --purge postgresql*" and reinstall postgresql-13. Now the output 
of "psql --version" is "psql (PostgreSQL) 13.4 (Ubuntu 13.4-1)". 
However, trying to enter psql, I have this error:

psql: error: could not connect to server: Aucun fichier ou dossier de ce 
type
     Is the server running locally and accepting
     connections on Unix domain socket "/var/run/postgresql/.s.PGSQL.5432"?

I assume this has been asked many times, but I tried different solutions 
proposed on similar posts and none solved my issue.

Thanks,

Best regards

Jeremy