If they say ISO is a component of exposure* then yes, they are wrong. The effect of ISO is applied after the shutter has closed and no more light can reach the sensor. So ISO cannot be a component of exposure*
For example:
For a given scene and lighting -
A. f/8, 1/400s, ISO 200
and
B. f/8, 1/200s, ISO 100
will both output the same image lightness but the ISO 200 shot will have had only half the exposure* of the ISO 100 shot and so will have more visible noise in the shadows because of the lower SNR resulting from the smaller exposure*
Exposure* and image lightness, although related, are two different things and not 2 different names for the one thing.
It's when people use exposure* and image lightness interchangeably as if they meant the same thing that they often get themselves in a twist, especially when I tell them I can change the image lightness in camera while keeping a constant exposure*
What is the definition of the word 'exposure' you use when you use it in sentences?
* exposure - amount of light striking the sensor per unit area while the shutter is open
** optimal exposure - the maximum exposure* within dof and motion blur requirements without clipping important highlights.
*** under exposed - more exposure* could have been added with the DOF and blur constraints still being met without clipping important highlights.