The software described would usually be classified as an Advanced Persistent Threat [1] or Rootkit [2]
Backdoor [3] usually refers to methods to sidestep authentication added by the vendor.
Many commenters are pointing out that one possible definition of a rootkit is something that elevates privilege, but does not necessarily have network communications functions or a command and control server. But in recent times, almost all modern rootkits seen in the wild have some form of network control functionality.
A rootkit isn't for privilege escalation - you need root before you can install the rootkit. This is typically obtained through a privilege escalating exploit, the rootkit is for maintaining access and masking the attack.
> The modified compiler would detect attempts to compile the Unix login command and generate altered code that would accept not only the user's correct password, but an additional "backdoor" password known to the attacker...This exploit was equivalent to a rootkit.
> Establish Foothold – plant remote administration software in victim's network, create net backdoors and tunnels allowing stealth access to its infrastructure.
> A backdoor is a method, often secret, of bypassing normal authentication in a product, computer system, cryptosystem or algorithm etc. Backdoors are often used for securing unauthorized remote access to a computer, or obtaining access to plaintext in cryptographic systems.
I read all of that as a backdoor being an umbrella term, of which one type is a rootkit, and APTs create backdoors, perhaps of a type other than rootkit (e.g. net backdoor).
The software described would usually be classified as an Advanced Persistent Threat [1] or Rootkit [2] Backdoor [3] usually refers to methods to sidestep authentication added by the vendor.