Hi Bas,
Glad that OneNote Online is working on the site collection created with the Project Site template.
So, the issue is related to the sub site of the particular site collection--root team site collection. Please allow me to clarify the situation again:
1. OneNote Online works in the root team site.
2. OneNote Online does not work properly in the sub site (root team site/projectsite1) that's created with the default Project Site template.
3. OneNote Online does not work properly in the sub site (root team site/projectsite2) that's created with a custom template.
4. Can you go to create a new sub site under the root Team Site with the default Project Site template?
And check to see if the Notebook works and create a new OneNote notebook in a document library to see if it can be saved. If the issue persists, provide the URL of this sub site and the screenshots.
5. Can you check to see if the issue only occurs to the sub sites created with the Project Site template?
6. Can you recall what you have done when the issue occurs or something related to the issue?
We'll involve high level members in to help look into the issue after we get the information above. Thank you for your understanding.
Regards,
Cherry