someone else's database that may or may not persist your data.
Discussion
Isn't that always the case?
Not if you use your own database.
Still no difference
Someone else's database where you have no control over whether they persist data vs your own database where you have full control over what data is persisted are totally different. Are you being deliberately obtuse?
A public nostr relay is akin to a public database with an API.
A personal private nostr relay is akin to a personal private database.
I fail to see any difference.