Rotor is the more generic term, every motor has a rotor, it's the rotating part relative to the mounting of the motor.
Both induction motors and brushed DC motors have "rotors".
As used by some, armature is a specific term that applies to some types of motors only. You can still call it "rotor" as well.
I find little use for the "armature" term because it's just a different name for the same thing that can be used in some cases, pretty confusing. I feel it easier to understand that in the typical brushed DC motor design, rotor just has windings and commutator copper segments. No need to give it a different name.
Wikipedia defines "armature" as: " the armature is the winding (or set of windings) of an electric machine which carries alternating current". This shows the bullcrappiness of the word "armature" - completely different meaning to what some use, and again it's a replicate. For windings, we already have the term: "winding".
So it seems "armature" can mean anything, and it's always just a duplicate name for something that already has a proper name.