Why S parameters are written as S(out,in): the first subscript is the port where the signal comes out and the second is the port where it went in. Forward gain means drive port 1 and measure what emerges at port 2, so it is S21. The reflection terms S11 and S22 have matching subscripts because the signal leaves the same port it entered.
Watch out S12 is the same path run backwards, signal in at port 2 and out at port 1, which describes reverse transmission or isolation, not forward gain.
Read S(out,in): forward means in at 1, out at 2, so 2 then 1 = S21. Matching digits = reflection.
Forward gain earns a 21 gun salute: S21. Matching digits like S11 and S22 are reflections, not gain.