How do I set up multiple sites with one database or multisite Apache configuration?
The new site vhost should be pointing to the drupal root installation where the index file lives and the entry point for all your sites, not to the site1 folder.
"DocumentRoot /var/www/multisite" this should be the same for all your sites.

With multi-site a approach you can shared modules between sites (those in all folder), have different modules per site, you can have different databases per site and share tables between them.

Your drupal sites are going to share the same code base, One drupal core for everyone.

This can be useful https://drupal.org/documentation/install/multi-site