OK I have tried your recommendations - I downloaded the files fresh, reset my site and loaded everything up again; and this time I gave the database and the user the same name. But, I am still having the same problem.
Here is the information that 000webhost gives me for my database:
I only named the database and the user 'puddles' - 000webhost adds the ID number at the beginning so I don't have a choice about that.
Here is how I am entering the information:
I have tried some different things and here were the results:
If I enter both user and database just as 'puddles' without the ID, it gives me a different error, it says I do not have permission to access the database as user 'puddles'.
If I enter user as 'id1039547_puddles' and database as 'puddles', it gives me the same original error ("SQLSTATE[HY000] [2002] No such file or directory"). So perhaps I am still using the wrong database name but I don't have any idea what it would be if it is not 'puddles' as I entered or 'id1039547_puddles' as they show me in the database information...