Mess with accounts in OneNote and OneDrive for Business - can't open documents from Office 365

Question

Hello!
[Reposted from
here]
I've got Office 365 subscription (E3) with Office.
I can view and edit OneNote notebooks on Office 365 via browser, but can't open it in OneNote.
The similar problem exists for OneDrive for Business - some libraries are syncing, some cannot - asking for Enter credentials, but when I click - nothing happens.
There were two accounts bound to Office 2013 - personal Microsoft ID as primary account and corporate Office 365 linked to it.
Windows 8.1 corporate profile is linked to Microsoft ID account as well.
Something happen with these accounts after password was changed - it seems like some account information was cached and can't be cleaned.
I tried to uninstall Office and install it - no success.
I cleaned computer with
this utility from Support - no luck.
On other Windows 8.1 computer I created new profile (with Microsoft ID account) - surprise, there are problems there as well.
I deleted all Office15 entries from Credentials Manager - Office asked me about Office 365 login/password - I entered them, it is syncing, Haliluja, but, when Microsoft ID account was linked to Office 365 account in Office - problem appears again.
It seems that Office 2013 uses Microsoft ID credentials for Office 365 content.
I'd like to use Microsoft ID as primary account for Office and link Office 365 account to it, but don't want such problems. It worked fine, but I don't know what happened and how it can be analyzed and fixed.
Update: It seems I localized the problem, but still don't know the solution.
Scenario to reproduce error for me:
1. Create fresh new Corporate/Domain user profile on Windows 8.1 with Office Pro 365 installed on computer with latest updates
2. Go to SharePoint Online and browse some Libraries and OneNote notebooks - all ok
3. Start to sync them using OneNote and OneDrive for Business - ok
4. Link Microsoft Account with Corporate Account in Windows 8.1 - ok
5. Syncing of _some_ (not all) Office 365 libraries and notebooks stop working - errors with access, permissions. I'm trying to enter credentials but this doesn't help
6. Disconnect Microsoft Account and Reboot - all libraries and notebooks of Office 365 work fine again. But OneNote notebooks from OneDrive stop working :)
How I can fix this situation and have Microsoft and Corporate account connected?

Answer 1

Update 2:
I installed PowerShell for SharePoint online cmdlets and found this user using get-SPOUser.
I removed this user using remove-SPOUser -site
https://mysitename.sharepoint.com -LoginName
[email protected]
but when I'm trying to access sharepoint from the browser with Microsoft account credentials - new record for this user is created again and I can see it with get-SPOUser :(
So, problem is still not solved - this user is still can be authenticated (there is no such user in SharePoint site collection, there is no such user profile) and Office 2013 syncing doesn't work for Office 365 due this mess with accounts.
I know that I can remove Sharing with external users from Site Collection and information about these users will be permanently deleted - but I don't want to do this, there are several customers who work with our documents.