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hi inovix
after i read the whole thread, i'd like to explain the following points:
1.the storage of the team site is absolutely not 1 tb. its' storage is 10gb+500mb*number of licensed users. refer to limits for sharepoint online in office 365 small business premium. for your convenience, see the screenshot below.
you can navigate to the user' team sites which url looks like https://contoso.sharepoint.com/sitepages/home.aspx. click the setting gear>site settings> storage metrics. the storage should be more than 10gb and less than 1tb.
2.the third picture on your second post displays 0.05gb which is a normal behavior. it is my site host's default storage. its url looks like https://contoso-my.sharepoint.com/_layouts/15/storman.aspx
3.the fourth picture on your second post displays 1024gb which is user's personal site (onedrive for business storage). you can log into the office 365 portal ->onedrive->the setting gear>site settings> storage metrics to check the outcome.
as you say "users get the alarm", since the storage of the team site only has 10gb+500mb*number of licensed users. i suppose it may approach the limit of the storage.you can let him:
1.upload files to onedrive for business which has 1 tb personal storage.
2.upload files to team sites, but they need to purchase addition storage.
here is a related thread for your reference: https://community.office365.com/en-us/f/148/t/203302
if anything is unclear, you can post back with the full screenshots including the url. we will help you understand these sites better.
regards,
sky