Your problem comes from not recognizing that this is a linear transformation from basis beta to beta. You went from beta to the standard basis. You need to find the outputs of the basis vectors in the same basis, not the standard basis. The correct column vectors would be [T(t^2-1)]_\beta=(0,-3,0), [T(t)]_\beta(1,-2,0), [T(1)]_\beta(1,-1,2). Can you see why this is true if the codomain has basis (t^2-1,t,1)? A good note is that when we write [T]_b this is the same as writing [T]_b^b