Hello - first off, take my reply with a grain of salt; I'm not a motor expert - my knowledge comes from just playing around with them...
1: I'm confused between the terms "rotor" and "armature". I think an "armature" is that part to which electric current is applied. For example, in an induction motor we have windings on both stator and rotor but electric current is applied only to the stator so in this case the stator would be termed armature. Is that correct?
The "rotor" is the part that "rotates" (notice the root/prefix of "rot"), the "stator" is the part that doesn't rotate - ie, it is "static" (notice the root/prefix of "stat"). As far as which is the "armature" and why? Hmm - based on this:
https://en.wikipedia.org/wiki/Armature_(electrical_engineering)
...It depends. Read the article for more info.
2: Many persons have commented that the motor shown in the
video is not an induction motor; someone has called it a "universal motor". What kind of motor is it really? I don't see any brushes etc.
No - you can't see any brushes, but you clearly see a commutator. My first guess would be that it is a universal (or something similar) brushed motor. Now - I suppose it might be possible to turn a universal motor into an induction motor if you shorted the poles of the rotor coils/commutator, then applied just the right waveform/frequency to the field coils...hmm, dunno.
but it still can't be an induction motor because there are only two poles (as far as I know an induction motor should have at least four poles). To help me with the query you only need to watch it from 00:30 to 01:22. I understand that the video is not a good one, I'm just curious.
From what I have seen of induction motors (firsthand, and reading) is that the rotor consists of special laminations (aluminum/steel/copper?) into which the stator coils "induce" current (like an electrical transformer) in and this current in turn is repelled/attracted by the field set up by the stator. The stator coils usually, as you have noted, are multiple; but I don't think the number has any real bearing, though you generally see more than two coils, simply because to induce rotation you can't have only two, unless you have an external means to "kick" the rotation going - also, see for example Michael Faraday's simple electric motor/generator, which technically was a form of induction motor with a single "coil" in the rotor, and an external permanent magnet field.
Generally though, modern induction motors have that somewhat "solid looking" laminated rotor, though some (I think) have a rotor with coils (it's just that the coils are shorted together in a certain manner; like I said, it may be possible to turn a universal motor into an induction motor in some manner - maybe - which may be why the person in the video refers to the motor as an induction motor, though it clearly has a segmented commutator at one end of the rotor).
Please keep in mind that I'm totally new to this machine stuff and in the past have only studied basic DC motor and generator so please excuse the complicated details. Thank you.
No problem - I'm fully expecting someone to come in here and kick me in the butt for leading you astray with incorrect information. Here's hoping we both learn a thing or two about these machines!