ddunk1946 on "WP 3.6 & database relationship..."

While not new to computers, I AM new to WordPress and MySQL. In fact, I first started to use WP a few weeks ago, when v3.6 was introduced. And I know that I will feel even dumber when I have the answer but here goes:
I don't understand the relationship between the db and WP. When 3.6 installs (1-click through my hosting company,HostGator) it installs a MySQL database called username_wrdp1. If my ultimate goal is to use WP for a web site (or two) and a blog (or two) why would you not rename (through PHPAdmin), the database something identifiable as belonging to that site or blog? Even updating the wp-config file to reflect the correct user name, password and database in question gives me a 'cannot connect to database'. I know it's 'pilot error' but I can't see it at the moment.I have been putzing with this for some time. I think I understand that 3.6 can do multiple sites/blogs, but each requires a different installation. That's not the issue at the moment but I don't want to spend time modifying a site and its database only to find I should have created separate ones. Is there a 25-word-or-less answer for this, or a link? Many thanks.