I want to implement a 2x2 optimal solver with SymCoords

[Renslay]

Nice. So compared to the standard definition you just have to add +1 mod 3 for the UBR position and +2 mod 3 for the UFL position, the other orientations are the same.

Not just that; it's a bit more complicated, but not much.

As for positions, whatever corner is at UBR position, I have to compare their reference sticker to R side. I have to read the stickers of the corners here in this order: RUB. Similarly, at UFL position I have to compare reference stickers to the F side: FLU reading order.
As for corners, the corner UBR has the reference sticker R (instead of U). Same applies to the corner UFL with reference sticker F.

So it's a bit more complicated, because there are 2 positions AND two corners that I have to handle differently.

Here is an example "read", after the random scramble R' F' U' F2 U2 R F' U2 F U' F.

XYZ is the reading order of the positions, and VWX is the corner at that position with the reference sticker W.

At position ULB is the corner BUL. Standard U/D stuff, orientation is 1.
At position RUB is the corner BDR. Now, BDR has a standard U/D sticker, normally this orientation would be 0, but comparing the the right side, it's orientation is 1.
At position URF is the corner FRD. Standard U/D stuff, orientation is 2.
At position FLU is the corner BRU. This is interesting, because I have to deviate in both the position and the corner itself from the standard U/D notation! I have to compare the R side of the corner to the F side of the cube: orientation is 1.
At position DRB is the corner FDL, standard, 1.
At position DFR is the corner FUR, standard, 1.
At position DLF is the corner LUF. While the position is standard, I have to use the F sticker of the corner instead of U: orientation is 2.

Overall, the orientation vector in this order is (1, 1, 2, 1, 1, 1, 2).
U/D version would be (1, 0, 2, 0, 1, 1, 1).

Note that there are 3 differences between the two vectors (marked in bold), because we have 2 special positions and 2 special corners, but here one special corner is in a special position (FLU position with BRU corner). So the difference between the two notations can be anything between 0 and 4 number of places.

BTW the solved cube in both notations has (0, 0, 0, 0, 0, 0, 0) as orientation.