Is there a name to this strange opening?

I have had a lot of recent success as black using the strange opening above, especially since it can lead to winning a piece with two different traps (variants in the board above). As far as I can tell, there has not been many games with it. Presumably, the opening is suspect, but stockfish rates this position a +0.30 which I feel is reasonable for black.

I know it starts off as Old Benoni, but basically, I skip NF6, in order to keep the option of playing Qf6 available. It plays a bit like a benko gambit, but to the best of my knowledge, it isn't actually a benko without Nf6.

How would you evaluate this opening, and why do people not play it more?

This is a bad idea to postpone Nf6 for two reasons :

a) White can try 3. Nc3 then Black can't take on d5 without creating a hole on d5 for the white knight. After 3. Nc3 if you play 3... Nf6 to exchange the N when it arrives on d5, then 4. e4 creates more problems as now 4... exd5 simply loses to 5. e5. The same happens after 3. c4 b5 4. cxb5 Nf6 5. Nc3 exd5 6. Nxd5 and Black runs into all kinds of problems (6... Bb7 7. e4!? and the like). And 4... a6 is a very poor Benkö-like as in the Benkö the bishop is really needed on g7 to pressure white's queenside.

These positions with a hole on d5 are fine if Black has enough pieces development to compensate for it, but it's not the case here.

b) after 3. c4 b5 White can simply play 4. e4 with quite a large advantage.

For all these reasons and more it has been established that the inclusion of c4 and Nf6 at move 2 is a general improvement for Black.

I believe it's called Horwitz Defense. But white cxb5 is obviously a mistake. Never take toward the side of the board unless there is a very good reason. White should respond with 4. e4 to form a strong pawn structure in the center. If black takes white c-pawn white has bxc5, threatening to trade blacks important e-pawn at some point.

I personally don't like Horwitz Defense because, if white is not familiar with this line, it can easily avoid it by doing 3. dxe6, and then trade the queen. White retains its positional as black has lost its castling. This line is my personal favorite playing white, because black's whole plan for opening will be nullified. Let's play the middle game!

As you mentioned, capturing to the centre is preferred. I always play 3. ...fxe6 here, and find it quite okay for black. There is no easy way for white to exploit that weak diagonal before black castles kingside. Plus, it opens up the F-file for my rook after kingside castling so I quite like this position as black actually.

If white goes 3.e4 then it’s a Franco Benoni (normally seen via 1.e4 e6 2.d4 c5 3.d5 move order) and white is clearly better. If White plays c4 or Nc3 on move 3 then they’re just better there as well. Personally the Franco-Benoni transposition is the main reason why I wouldn’t consider this but even the other lines just seem like a worse version of an already risky opening (I don’t see the value of not developing the N to have Qf6 available)

Isn’t that old benoni defense

I agree black's position is better if playing 3...fxe6, but still what is black's plan? You mentioned castle king side, but it is not possible because you c-pawn blocks you dark bishop, and since f-file is open, g6 is not an option. So where will your dark bishop go?

I know that engine says it is a equal game , but for human players it is a very awkward position for black to maneuver.

If black forces a king side castle by playing 4....be7; 5, nf3, nf6; white has e6 which kick black knight and winning tempos in development.

The best strategy for black is to not seek castling, and develop its queen side instead. But again, that will be an awkward position for human to navigate.

Usually, I go Ne7 Ng6 to dodge e5, then put the bishop on e7. it's actually quite a comfortable position for black to play personally, but I have played this line multiple times.

Yeah it is, sort of. It's a weird Benko without the Nf6. But as some people above pointed out, an early Nf6 is important. Not gonna lie though, at my level, this variant has served me quite well, netting a number of early wins.

I see you point. Personally i think it's not good for black. You know the infamous Cow Opening invented by Anna Cramling? Your knight takes the same moves as her cow opening.

GMs studied Anna's cow opening and concluded the best attack against cow is to push your a or h pawn to threat its knights. So, in you situation, I would do the same. See below.

I think black is in trouble.

I think it is quite silly to play be7 here, obviously you need to play Nc6 here first. The point I was making still stands. As long as you play Nc6, D5, and Be7, in some order, you're probably fine; it doesn't really matter whether you place the knight on f6 or g6