Your 'normal' exposure index of a film becomes an over-exposure or under-exposure in lighting conditions other than 'normal'.
If it's foggy, for example, or heavily overcast, you usually have much less contrast in the scene. This means you can expose your film at a higher exposure index than 'normal' and still retain full detail, because the brightness in the scene is compressed. By underexposing you stretch the shadows back down towards the toe of the film curve, and then you over-develop the film, to stretch the highlights up the curve to make a negative of 'normal' contrast.
I would not call that push processing. I would call it underexposure and over-developing for normal contrast. Push processing usually involves photographing in very low light where you don't have enough film speed to record the whole tonal range, so you sacrifice shadow detail in order to get the rest of the tonal range back into their normal places. The shadow detail is the main distinction here. In low contrast photography you don't push process, you just compensate for the low contrast of the light you're photographing.
If you have very bright late afternoon sun, with super bright highlights, and very long deep shadows, to generate a 'normal' negative, you would have to give plenty of exposure to record what's in those deeeep shadows, and then to avoid blocking up the highlights you process for less amount of time. This is also a way to create a normal contrast negative, and in my mind pretty standard practice in making sure I have printable negatives that don't break my heart at printing time.
Knowing what to expect is key, in my book. So underexposure in low contrast weather can be as much as two stops to yield a normal negative. Over-exposure in extreme contrast can be as much as two or three stops, just to capture all that shadow detail. To me that is just a variation of 'normal', because that's what I aim for the negative to be.
With that said, you will find people (myself included) who think that an ISO 400 film like T-Max 400 looks best shot at 1600 in normal lighting conditions, just because we're not terribly concerned with shadow detail, and may actually prefer that there is NOT shadow detail all over the place. That is a personal choice, according to preference and taste, and nobody can argue that this is wrong either. The point is that you do with your negatives what looks good in the prints. Shoot a roll and bracket your exposures, see where you get 'enough' shadow detail to satisfy your taste. Then adjust developing time until the rest of the tonal range looks right.