but how can I hide avatar URL on the registration page?

Traits are designed to be supplied be the user during registration. You can hide them though if you use a custom UI and filter them out.

so the correct way to store the avatar URL would be metadata_public?

That's probably what you want - or store it completely outside of Ory since it's not really relevant for Auth, you can use the identity id or identifier for association

that's interesting, like fourth way to store metadata

see here also https://www.ory.com/docs/kratos/manage-identities/best-practices#keep-your-data-lean