Names of NAD+ and NADH As part of the biochemical process of glycolysis, NAD+ is reduced to NADH. In other words, nicotinamide adenine dinucleotide is reduced to... nicotinamide adenine dinucleotide?

Everywhere I look tells me that NAD+ and NADH refer to the same particle with the same name, just in an oxidized or reduced form. However, NADH has one more hydrogen atom then NAD+, so shouldn't these particles have different names?
The H stands for hydride, but most everyone just shortens nicotinamide adenine dinucleotide hydride to NADH.