My understanding is that the term "network" is misleadingly used by WordPress to mean the same thing as multisite.
Close. It's not misleadingly :)
WordPress Multisite allows you to create a network of sites out of one WordPress installation.
Multisite's the name, Network is the 'thing.' You're the 'Network Admin', and the pages are wp-admin/network
See, we have a weird split decision about 'site' vs 'blog', and what does 'multisite' mean and ... blah blah blah. It's not MEANT to confuse people, but it does. WordPressMU (the precursor to Multisite) called everything blogs. Multisite we call them 'sites' (hence MultiSITE).
And you can have MULTIPLE networks in ONE Multisite install. Just to throw more eggs in your basket.
Yaaaay :/
In all cases: The sites aren't shared. They can be with plugins, but by their nature, they are multiple, separate, sites all using the same network. See wordpress.com :)
sp12, the tl;dr for you is this: Install WordPress. Activate Multisite. Configure your network.
If you want all users to have access to all sites, we have plugins for that:
Automatically add new users to each site in your WordPress network
* http://wordpress.org/extend/plugins/multisite-user-management