The first option you discussed is a lot more common - the moodle site sits in a subdomain behind the main website. It is very easy these days to skin both site to look similar (I have done it before using a Wordpress/Moodle combination).
There is actually no real reason technically however why you can't create your entire website within the moodle installation. You can skin moodle to make it presentable and then use custom navigation and html labels to give it more of the "website" feel.
The drawback you have to consider though is the limitations you will have doing it this way. Moodle is a Learning Management System, not a Content Management System. Moodle will not give you a lot of freedom when creating your new site whereas using something such as Wordpress or Joomla will give you much more power to manage and manipulate the way visitors access your information.
Put simply, if the "website" component of your project contains only a few pages of information then go for it, but if you have a complicated plan stick to the two site system.
All that said, I have never seriously tried to create a website entirely inside moodle and the neatness of the proposal really appeals to me so it is certainly something worth developing to see what could be done