Is it NAD+ or NAD- or neutral NAD ?
You are correct in thinking that because of the presence of the diphosphate linkage between the two nucleotides in NAD⁺, both of which will carry a negative charge at physiological pH, together with the positive charge on the nicotinamide N, it would be more chemically correct to write NAD⁻. However I think the idea of using NAD⁺ is to emphasise the N atom in the nicotinamide moiety, which is the 'active site' of the coenzyme.

As it says at the NAD⁺ Wikipedia page: 


  Although NAD⁺ is written with a superscript plus sign because of the formal charge on a particular nitrogen atom, at physiological pH for the most part it is actually a singly charged anion (charge of minus 1), while NADH is a doubly charged anion.