The way I see it, is that for a rook, it is easy to know which path it can follow. You simply take it's current position and allow it to move to any neighbor tile sharing an edge, and then the opposite tile, and the opposite again, and so on.

For a pawn, it can also move to neighbor tile sharing an edge, but there are multiple ones, and only one is allowed. So how to you define which it is?

If we were to say, as I understand is what you suggest, that it will follow the same path as a rook, here is the problem that I see. If it was in one of those tiles you mention, we can't infer which path a rook would take from the pawn current position. We would instead need to know its starting position. And by just looking at the board in a given state, the pawn could come from 2 places ( see https://imgur.com/a/sExTqsP ). So that would require the player to always keep in mind which pawn comes from where. Which I think is really hard. Especially for games with no timer, as those could last days.

I'm much more in favor of a definition of forward which is comprehensible by just looking at the board, without having to think about the starting point. Which is why for now I've settled on using the direction of the tile edge. Even if there is a non-negligible cost of visual clutter.

Also, I'm quite keen on preserving the fact that all pawn can be promoted to a piece when reaching the opposite edge. As soon as you know that a pawn can't do that, they lose a lot of their threat potential.

But I don't want to close off this idea. It does make me think, and I'd be happy to be convinced otherwise. And it could very well end up as a variant inside this variant!