We get a clue by looking at the path in the error (we know it's linux) and from another clue we see CPanel (remotely hosted with a provider but which) and there is mention of a fact you've been successful installing Word Presses and Joomlas.
If the database server is on same server as the web server, php and Moodle code one would use 'localhost' for the server to which to connect. If the package you've purchased offers MySQL but on dedicated DB servers, that info should show in CPanel and you'd be using an IP address.
Think I'd look at config files for Word Press or Joomlas you've installed on similar provider servers to see how the DB is configured for those apps for clues on how it should look in the config for Moodle.
There is also a hint that one is accessing the DB server with a user (moodle - in your second posting of debug info) for the MySQL server on localhost using a password, but does the DB server's user table have a user called 'moodle' and does that user have permissions/access rights enough to do pretty much everything with the DB setup for the Moodle?
'spirit of sharing', Ken