I think you are spot on for rootkit, but you are absolutely wrong on backdoor.

You say "This conversation is the best possible example of why we can't allow the corruption of previously well defined words - it causes confusion for no good reason." when YOU(and others like you) are the one corrupting the meaning of backdoor.

Backdoor has meant for ages to be a way to access a computer/program while bypassing the normal authentication method, whether added by the designer or by someone else. You are trying to redefine it to mean only methods of bypassing normal authentication added by the designer. If you find it confusing that both types of backdoor are backdoors, then make up a new word that can be considered a subtype of backdoor don't try to coop an existing word and change its meaning.