In Shogi and Shogi variants, they have the pawn, Iron General and Stone General.
They're equivalent to the pieces in your chart:
Shogi pawn --> Soldier
Stone General --> Stone General
Iron General --> Sergeant
https://en.wikipedia.org/wiki/Dai_shogi
I'm not sure if 'asymmetrical pawns' exist, so diagonally forward but only left, or only right.
After thorough research (and by that I mean 3-4 minutes of scribbling in a note-taking program), I have discovered that there are exactly 9 possible combinations of a Pawn variant moving forward. capturing forward, moving diagonally forward, and capturing diagonally forward. Out of these 9, Chess.com has 5 of these possible Pawns available. Here's what I found. There should be a chart at the bottom of the post if you're interested.
Firstly, Chess.com already has:
Pawn==Moves forwards, captures diagonally. Soldier==Moves and captures forwards. Berolina==Moves diagonally, captures forwards. Stone General==Moves and captures diagonally. Sergeant==Moves and captures in any direction.
With those out of the way, now we can talk about the new Pawn variants (comment name suggestions if you have any).
First up, a Pawn that moves forward and captures both forward and diagonally. This one is essentially the same as a normal Pawn, but captures forward. This one would be pretty formidable, especially in the opening, where it could grab more center space. You cannot simply place a piece in front of this Pawn to stop it from moving. It could also make some deadly and mostly impenetrable Pawn-walls that can't be bypassed by something like a sidestepping Knight.
Second, a Pawn that moves diagonally and captures both forward and diagonally. I can see it being really easy to blunder a Queen or trap a Bishop with this Pawn on the board. I could also maybe see this Pawn using discovered checks to it's advantage to scoop up some free pieces. It seems very chaotic. The diagonal movement can potentially be confusing, however.
Next, a Pawn that moves forward and diagonally but captures forward. This feels like a very aggressive, in-your-face-style Pawn that could maybe work well with Knights (I'm a 200 ELO, that might not be good advice). However, I think you would almost have to play offensively with these Pawns, since they can't exactly form Pawn chains, and they have a less restrictive scope. a Bishop might be this Pawn's worst nightmare.
Finally, a Pawn that moves forward and diagonally, and captures diagonally. This is obviously a stronger version of a normal Pawn. I could see it being utilized in a variant like Petrified (or just a closed position in normal chess), where you have obstacles that stop normal Pawns. I think this pawn would really shine in endgames, because stopping this Pawn from promoting is an absolute task. You have to account for 2-3 possible advancing moves instead of just 1 for a normal Pawn, essentially forcing you to stop them with a Rook or a Queen, which isolates the Rook or Queen from whatever else is happening in the game.
I hope you enjoyed reading this. Comment your ideas for names if you have any. If you want to concept art any of these, knock yourself out. Let me know any other thoughts you have about the strategy that could go with these Pawns.
-Sinoceratops317