Your answer is correct. Note there is no unique way to write a recursive formula for a given sequence.

To see that both recursive formulas coincide (yours and the one in the solution): assume $2\cdot 3^n + a_n = 3 \cdot a_n + 2$, then $$\begin{aligned}2\cdot 3^n + a_n &= 3 \cdot a_n + 2 \\ 2\cdot a_n + 2 &= 2\cdot 3^n \\  a_n + 1 &= 3^n \\ a_n &= 3^n -1.\end{aligned}$$ Note we know this last equation is true, since it is the answer for part (a), so both formulas determine the exact same sequence.

Also, remember that when giving a recursive formula you must also provide the first value of the sequence.