The professor in that video is trying to demonstrate the effect of feed back on noise and other non-linearities.
Lets start by assuming the amplifier is ideal: infinite gain, infinite bandwidth and capable of driving any load. I know these do not exist but it makes it easier for us to do a thought experiment.
Taking the circuit of Figure 1 then the system always has unity gain but we have a source of noise and non-linearity. The amplifier is forcing the voltage at the wiper of the potentiometer to match \$ v_i \$. If we imagine the wiper fully on the right hand side as drawn then the amplifier sees the effects of noise and non-linearity and corrects for it totally, With the wiper fully on the left hand side the amplifier does not see any of these effects so there is no correction and we see the full effects of non-linearity and noise. With the wiper somewhere in-between there is partial correction.
To clarify the point regarding always having unity gain I mean unity gain with some superimposed distortion resulting from noise and non-linearity. The position of the potentiometer controls the amount of distortion seen.
With your circuit of figure two the amplifier sees all the effects of non-linearity and noise so these are totally corrected for. However this circuit also introduces gain. The larger \$ R_1 \$ or the smaller \$ R_2 \$ the more gain and.
$$v_o = v_i \left(1+\frac{R_1}{R_2}\right)$$
