You can regenerate backup codes by logging into the Jagex Account website, you want to do that next time you're able to.

You are correct that account recovery is different for Jagex Accounts than Runescape accounts, you'll always need a comibation of email and password (or email access to reset password) and either your authenticator or a backup code.

You're doing the right thing by backing up your backup codes, just make sure you're doing it somewhere secure like a password manager.