Not quite. Chaturanga allows a pawn to promote to the piece whose starting
square it reaches--a pawn prmoting on a8 becomes a Rook, on b8 a Knight.
It doesn't matter which pawn it is, only which square it promotes on.
In Tamerlane's, the Rook's pawn always promotes to Rook no matter where
on the back rank it promotes, the Knight's pawn promotes to Knight, etc.
Here what square the pawn promotes on doesn't matter and which pawn it is
does--pretty much the exact opposite of Chaturanga.