The states which have a classical limit are coherent states which for each mode $k$ are of the form $|z\rangle =e^{-|z|^2/2}e^{z a_k^\dagger}|0\rangle$. These have $$ \langle z |a_k|z\rangle = z. $$

The states which have a classical limit are coherent states which, for each mode $k$, are of the form $|z\rangle =e^{-|z|^2/2}e^{z \hat a_k^\dagger}|0\rangle$. These have $$ \hat a_k|z\rangle= z|z\rangle, $$ and hence $$ \langle z |\hat a_k|z\rangle = z. $$