In Wikipedia, there's an exact solution of Discretization , which says the state-space model
$$ \dot{x} = Ax(t) + Bu(t) $$
can be discretized to
$$ x[k+1] = e^{AT}x[k] + A^{-1} (e^{AT}-I)Bu[k] $$
as an exact solution for this problem. Then, it says that applying the approximation $ e^{AT} = (e^{-AT})^{-1} \approx (I-AT)^{-1} $ , this solution becomes identical to the one derived with backward Euler method.
However, this approximation leads to a different result with the approximation $ x[k+1] \approx x[k] + T(Ax[k+1] + Bu[k+1]) $ . The first one, Wikipedia version, leads
$$ x[k+1] = A_dx[k] + B_du[k] $$
but the latter (which I believe is correct)
$$ x[k+1] = A_dx[k] + B_du[k+1] $$
for $ A_d = (I-AT)^{-1} $ and $ B_d = (I-AT)^{-1}BT $.
What am I missing in the first approximation?