As Jim said...
Or to put it more plainly, if there is more than one way to do it, you don't 'have to' do it one way.
Or a more 'pedantic' answer. 'pushing in PP' is not changing ISO, since ISO is a guide for setting exposure. What it's doing is changing the lightness for a given exposure, which the ISO control on a camera also does.
So what is wrong with exposure triangle which says that three components (ISO, shutter speed and aperture) regulates the amount of light that hits the sensor. Yes, you can change ISO on camera or push it in PP. But if you are using Auto ISO on camera, changing SS or F stop, changes ISO.
Is it bingo yet? Because that's what I meant to ask about...
If an in-camera push development of a raw file is not equivalent to an increase of ISO, except for resultant image brightness, then what is the exact difference between the two manoeuvres?
Refer to my more pedantic answer. 'Pushing it' in PP isn't really changing the ISO, since ISO is a guide to setting exposure.
Given a lot of surrounding explanation, the triangle might engender fewer conceptual errors, but I've never seen a version of the triangle which includes the explanation that you gave above.
Changing ISO doesn't necessarily change the amount of light that hits the sensor. Aim your camera at a scene. Set ISO to 100, shutter to 1/100th, f-stop to 8, and don't change the scene lighting. Now change the ISO to 200. The same amount of light hits the sensor.
Sensors clip hard. Changing the mis-named "Exposure" control in Lr introduces a shoulder, kinda like film. In some cases, increasing ISO can result in a decrease in read noise, especially in dual conversion gain cameras. Did you read the Lensrentals blog posts that I linked to above?
That depends. ISO only defines the relationship between exposure and lightness for in-camera processing. So, it's not strictly correct to say that changing the lightness in out-of-camera processing is changing the ISO - it's just processing as you want.
The ISO control itself likely has a number of un-mandated side effects, designed by the manufacturer to optimise for the expected exposure at that ISO setting. Being un-mandated these are completely unstandardised and left to the design of the individual camera. As a rule they are designed to maximise saturation capacity at low ISO settings and minimise electronic noise at high ISO settings. But if you know your camera well you might find a better combination of exposure and ISO setting than the 'default' ones - assuming that you do the processing out of camera (or change the defaults for in-camera processing).
The shutter speed and aperture (and the brightness of the scene) determine the amount of light hitting the sensor -- i.e. the exposure. Changing the ISO number, if you are in A, S or P mode, will put a bias on the metering.