$\overline A$ is the set of limits of convergent nets with values in $A$
Good for mentioning the rôle of AC. Explain why $\mathcal N_x$ is directed under $\le$ (yes, intersection of neighbourhoods ...). Add also the argument why $(x_U)_{U \in \mathcal N_x}$ converges to $x$ (it's a minor thing but shows why we have to define the pre-order the way we do). It depends on the intended audience, I suppose, but in home work or self-study fill in all the details...
The reverse direction I have no quarrel with, as remarking that $x_{d'} \in A \cap U$ to witness the non-emptyness, might be too trivial? I'll let it pass...
Ideas are fine.