To be honest, I think you are over thinking this and worrying about the how's and why's too much. It's making this, for your purposes, far more confusing than it needs to be.
To put it as short and simply as I can: The process of Syncing data is simply the automation of moving data between different devices. This relies on your account to determine where the data is coming from and needs to go. Setting up the app on multiple devices and signing in with your account will allow data to automatically be transferred between the devices.
In a bit more detail (although it's getting a bit more complex for what you are after)
A basic definition of the terms below:
Account: The account is the... account. Same principle as say your Bank Account. It's your identifying information with the provider, and is where all your data and files end up when sync'd.
App: This is the Application or Service itself. This can be installed onto your device, or a web server accessed through a browser.
Cloud: Is a buzz word. It simply means a remote storage location. They will usually come with an App that allows data to sync into them.
Sync: This is the action of 2 sources sharing information. It can be 1 way (one source needs to be the Master and sends all data to the other. Changes in the second source won't be reflected in the second) or 2 way (data travels in both directions so changes to either source will appear on the other). Most Sync's these days will be 2 way.
For a sync to work you will need to have an account setup for the data to sync into.
The main principle of Syncing is that data is moved between devices automatically via the internet. In order for this to occur an account is required to identify you and your data and for a location to be saved into. This can be at a Operating System and File level (such as OneDrive and iCloud, in which the buzzword of 'Cloud' is often attached), or can be at an App level.
So, in order to envisage how all this works, picture a wheel. In the centre you have the Cloud storage location, which is within your account. It is the Master source for the data. Each spoke is effectively a App Location that syncs to and from the center. There are as many spokes as you have app connections (which is often limited by license restrictions in the app, but that's another discussion).
As a change is made to the data on one of the spokes, the sync process on that device then automatically transfers the data files (or the delta between original and new) into the central cloud location. Each of the spokes' sync services are also automatically (be it constantly, on a minutes/hours/days, or on a button press in the app basis) polling the center to see if there are any updates to the files. As a change to the center is detected, the files are downloaded by each of the spokes.
So how do all these tying in together?
You want to use OneNote. You can use it in 2 ways:
Offline/No Account: You can use it offline without the internet, and without an
account. Your Notebooks are stored locally on your device. If you want to use these notebooks on a second device, or back them up, you will have to manually move the data files between the devices yourself.
Online / with Account: You can also use OneNote online with an
account. The
account will identify you and your data and will allow the
sync to occur. The
sync basically just moves the data files automatically into a
cloud location. This then allows the
sync to occur on other devices where you are signed into your
account and keeps everything updated. This
sync to the
cloud is only for the
app data and doesn't include other data outside of the
app.
So the
sync is basically just the automation of the data file move that you would otherwise be doing yourself. It stores these files in a remote storage location (which is often refer to as
cloud storage) as the central repository for all devices to sync to and from. There are some smarts around how to handle concurrency and conflicts, but it's mainly just about automating what used to be a manual process of moving data between locations, and having it backed up and available from a central location.