fix(config): use auth_db as database name

[MarkCesium]

Summary

.env currently points to placebrain_auth. The infra side is unifying on {service}_db naming (matches .env.example here and existing init scripts for other services). Align.

Changes

• Update .env DATABASE__URL database name to auth_db
• No .env.example change needed — already correct

Migration note

On developer machines with an existing var/postgresql volume and data in placebrain_auth, either:

• Dump/restore into auth_db, or
• docker compose exec postgres psql -U backend -c "ALTER DATABASE placebrain_auth RENAME TO auth_db;"

Blocked by

• fix(postgres): add auth_db init script and unify DB naming infra#5: add init-auth-db.sh and unify naming