When to choose pg_dump, pg_restore or psql to backup and restore Postgresql database?

If you created an SQL-format dump, all you can use is psql.

If you created a custom-format (pg_dump -Fc) or directory-format (pg_dump -Fd) dump, you can and must use pg_restore.

Custom and directory format dumps offer a lot of advantages over plain SQL script dumps, and I use them exclusively. You can selectively restore only some tables/schema, can choose whether to include only schema, only data, or both at restore time, etc. Lots of the options you have to specify at pg_dump time with SQL-format dumps can be chosen at restore-time if you use a custom-format dump and pg_restore.

If it weren’t for backward compatibility I’m sure the default for pg_dump would be -Fc (custom) format.

You can’t convert an SQL-format dump into custom-format or directory-format without restoring it to a PostgreSQL database and then dumping the restored DB.