Originally, OneNote notebooks have been stored to local drives only. OneNote 2007 added the option to use network shares; SharePoint and Microsofts free cloud storage service (originally named “Windows Live Folders”, then “SkyDrive”, now “OneDrive”) followed later.
After Microsoft’s change of course in 2014 (“Cloud First Mobile First”) almost all new OneNote versions (mobile apps, MacOS, OneNote 2016 free version, OneNote app for Windows 8, UWP app for Windows 10) have been limited to OneDrive and OneDrive for Business as a storage location. Thus, a free Microsoft account or an Office 365 subscription is mandatory to use OneNote at all. Only the “full” version of OneNote 2016 (and it’s predecessors OneNote 2003-2013) are still offering the option to save notebooks locally. “Saving locally” has nothing to do with the local notebook cache which exists for every OneNote version and allows you to temporarily work offline until the notes get synced to the cloud when the connection is re-established.