Right, but the "get new key" bit means that your accounts are in a vulnerable state while you're getting the new key.
If you have two independent keys, and you learn that your backup key is compromised, you immediately revoke it with all services, and order a new one. When the new one arrives, it becomes your new backup, and you enroll that in everything. Your vulnerability to an attacker ends immediately after you find that your backup key has been stolen and you revoke it.
If you have two identical keys, and you learn your backup key is compromised, you order a new one (or pair, rather), but you can't revoke the old key until the new one arrives, when and you can (simultaneously) revoke the old and enroll the new.