The private account is the one created by you.
If anyone knows the exact difference, please explain! Would be greatly appreciated.
The Personal accounts (also called Microsoft accounts/IDs or Live IDs) are the ones you as an individual create directly with Microsoft. The prime example would be people who've had a Hotmail account for ages, and that eventually became a Microsoft account.
The confusion comes because since a couple years ago, you can create Microsoft accounts with any e-mail that you own, be it personal, from work, hosted in gmail, hosted by your employer, hosted by yourself... You name it. So it is possible to create a Microsoft account with your work e-mail, even if your work e-mail also has a Work/School Account tied to it. All the notifications for that Microsoft account will go there, but they are not for your "Work/School Account" (also referred to as Organizational account), they are for your personal one.
Yes, it is a mess. After a while I got used to it and now I'm comfortable navigating all my identities, but it can be very annoying when you first encounter it.
Gawd.. You're microsoft, you can create a whole new security protocol. Don't have this half baked persona solution.
If you can't figure this out, it really reinforces the reason for every time we come across something in Azure where we're like wtf isn't this feature supported?
But, ultimately, those licensing rules may be under MS's control as well. Even if they aren't, it sounds like it has a significant impact on their service's design and usability, so it's in their interest to make a change somewhere.
Just merge them into a single collection and have my experience be dictated by that merged collection. With how entrenched they are with the Enterprise world, they should have foreseen this scenario and designed their family of products to facilitate this seamless merging of authorization rules across disparate accounts.
Because user stories aren't: "I want to access my Office365 work account" or "I want to access my personal Microsoft account".
They're: "I want to edit or download a file I have access to in the Microsoft cloud" or "I want to find a file I have access to".
The school/work account is somehow created by a company administrator, but I don't know exactly how or where.