The two terms overlap, but they are not identical.
On-premise AI
The software and models run on servers you own and operate, usually in your own data center.
Self-hosted AI
You run the software yourself instead of using a vendor’s hosted service. That can be your own data center or your own cloud account.
Which to choose
If strict data residency or isolation is a requirement, on-premise or air-gapped setups give the most control. If you want control without managing hardware, self-hosting in a private cloud account is often a practical middle ground.