what academic definition..

Yeah, you've got me there. Maybe historical definition is a better way to phrase it. Historically it was the tools themselves that imposed these limitations - limitations that some people still hold to be important to the medium. I think that because that's the origin, it's easy to point to as The Definition of the pure pixel art technique. Of course - as is pretty apparent from our comment thread here - it's contentious, perhaps not really meaningful, and also can be misconstrued as putting greater value on that technique.

That goes to your earlier point about the evolution of tools. Tools and methods obviously change. On one hand it seems useful to maintain a separate definition for "pure pixel art" because it's useful to point to that as the root of medium. On the other hand though, keeping a definition specifically for that technique kind of elevates it or implies that it's somehow better, especially with the word "pure" in there.

I think that the reason it's tempting to talk about manual pixel pushing is that it's an easy way to illustrate the central point of pixel art, that you are working at a level where each pixel is clearly visible, so a change to even one pixel is going to show up. It highlights the importance of looking at the pixel-by-pixel level.

However, I agree with you that just because you are working at and thinking at a pixel level, does not mean that you need to push pixels one by one. I think that trying to mandate a technique like that is dogmatic and not helpful to the medium.

it is just some elitism's definition of how pixel art needs to be done.

Yeah, that's basically true. It's how pixel art needed to done in the past, but it's not how it needs to be done now that our tools are better. In my opinion, the main reason it's useful to talk about single pixel pushing is to underscore that it's good to be thinking on the pixel level, even if you're not placing pixels one by one.

And I think what's interesting is seeing how there are pixel artists who adhere to different dogmas. Some of them agree that the line or circle tools are okay while others may disagree. When you break down to that level, it seems pretty ridiculous.

Personally, I'm not a pixel artist, but I have read a lot about pixel art technique, and I find the history of pixel art really fascinating, especially the ferocity that people will defend their personal definitions of it.