If you choose the on-premise options no data will ever leave your company. Administrators will install the software on a server inside your company and will be responsible for updates, backups and security of the server.
Still, again, as a power user... native applications are hard to beat.
‘Hosted’, to me at least, means that you’ve got an augmented web server somewhere on your intranet that you connect to, and that displays its interface through a browser on the user’s machine.
The two terms are mutually exclusive, in my understanding (except of course when one is speaking explicitly of “hosting a web server”, but I’m sure you catch my drift).