Seems like literally any cloud hosting at all is considered third party by them, which would limit schools to running LAN based apps only with all local or LAN

Seems like literally any cloud hosting at all is considered third party by them, which would limit schools to running LAN based apps only with all local or LAN storage. They have an overly broad definition of 'third party', because the usual concerns are data being leaked and used for advertising, not hosting servers, databases, disks, or other app services, which usually have some privacy terms. Some people might think a non-cloud based solution for a school would be better, but would almost certainly be worse. Schools are terrible at IT, security, updates, provisioning, etc and very quickly, it would be hacked, if not by external people, than by one of their own students.