Hi David,
According to your description, I have known that you would like to remove all traces of Office 365 to prevent a conflict when you reinstall Office 2010. If I have misunderstanding, please correct me.
The traces in the cloud would not affect the installation of Office 2010, however, the Office subscription may have some impact on installing the Office 2010.
1.For PC: To uninstall the Office 2013 completely, please refer to this article: Uninstall Microsoft Office 2013 or Office 365.
2.For Mac: It is not that complicated as for PC, you just need to uninstall the Office 2011. Then you can reinstall the old Office version successfully.
3.Office 365 is a set of productivity and workplace collaboration tools delivered through the cloud. Any changes in Office 365 are saved in the cloud rather than in the local computer. Therefore, it won’t make any problems when you install some softwares on your computer.
If there is error message when you install the Office 2010, please post back here and we will try our best to provide assistance.
Thanks,
Tracey Wang