newtonian mechanics - When the oscillator is a system with an angle can we define the angular frequency to be the radians per unit time covered by the system itself?

I read on stackechange that in springs or any one dimensional oscillator the angular frequency is just describing a rate of angle change in the associated circle on which it's projected. Something like this :

My question is: suppose you have a pendulum as an oscillator. Would it be correct to say that the omega / angular frequency. Is a measure of radians per unit time that the pendulum itself going through. Or there is still some other hidden circle for which this is defined?

Edit: is the following definition possible? : we take the whole edge on which the pendulum is passing through and circulate it. Meaning, we make a closed circle out of it . Would then the radians in that circle can be thought of as the angular frequency? If it's correct, would that be correct for the line on which a linear spring is oscillating?