A backdoor doesn't need to be remote and the user isn't necessarily an attacker. It is simply a secret method of access that the designer put in place, it isn't designed for end-user use. It is almost always security through obscurity, and it is always a bad idea. It can be activated in a variety of ways: port knocking, hardcoded passwords, preinstalled remote software, shorting ground to some magic pin, an undocumented serial terminal, etc.
A rootkit doesn't need to be remote and the user isn't necessarily an attacker. It doesn't need to have any functionality for user interaction - which means no "escalation" occurs (It could simply scan memory for passwords and log them to a file). It runs above user space, and can therefor be completely hidden (but it isn't always, see DTrace). It runs with the same privileges as the OS that it is part of. That is important to keep in mind, the rootkit becomes part of the running OS - that could mean any of the OSes running in your tower (CPU, HD firmware, BIOS, etc).
Your definitions work fine in a vacuum, but they quickly fall apart in real world usage. For example, by your definition: a remotely accessible privileged service is a rootkit, because an unprivileged internet user can interact with it - accessing data and executing code in the service's privileged context. 'sudo nginx' is not a rootkit.