Film's ISO speed is a property of the film and does not change no matter how you expose or process it. (One might argue some films are more honestly rated by manufacturers than others, but this is another issue.) Pushing films is merely underexposing and boosting the contrast by overdeveloping to make the image printable. As a direct consequence of this, some shadow detail is inadequately exposed to register anything on the film when underexposed, and this loss cannot be salvaged by overdeveloping. Grain increase is a secondary consequence of overdevelopment.(posted 8346 days ago)Some developers such as Ilford Microphen and Eastman Kodak T-MAX developers can increase usable speed by about 1/2 stop. This is a boost in true usable speed (but not increase in ISO speed - ISO speed is fixed) and not a push. This is because these developers develop more shadow images than the developer used in ISO standard measurement while contrast and speed point criteria are held fixed. Perhaps you need to find a carefully written tutorial article to learn more detail on this.
There are three things (or possibly more) for each film: (1) ISO speed, (2) effective or usable speed for particular processing, and (3) exposure index actually used for metering and exposure.
(2) can be different from (1) but usually this does not mean push or pull. However, if you choose (3) different from (2) then it is either push or pull.
My personal opinion is that push/pull isn't worth spending time unless you have a very good reason to do. Those techniques were probably more useful when papers didn't come in many grades or extreme grades didn't have equal image quality as grade 2.