It is important to distinguish true hydrous minerals, such as serpentines, from minerals which bear hydrogen as a defect, such as wadsleyite. NAM is a specific, well-defined term commonly used in literature which certainly applies to wadsleyite and ringwoodite. DMHS is not so common (perhaps you mean DHMS, dense hydrous magnesium silicates?) but should strictly refer to true hydrous phases such as Phase D.
Important is also if the mineral has avrying capacity to absorb water as opposed to is water formsa distinguishing parte o it structure. This acan be tricky i some cases as water is often the first thing to go in chemical analysis. If it is important to the structure then in does not matter how much water it contains. As you know water refers to either hydroxyl or actual water content. In wadsleyite and ringwoodite for example water is not needed for the structure as a defining attribute. So as Susannah stae any presence would be as an imurity or "defect". Hence, this makes NAM the term as either mineral are not hydrous - i.e. are by definition anhydrous.
your co-author is right. Both DHMS and NAM are dense Mg-rich silicates (they are stable in a Mg-rich peridotitic matrix), but the first are, by definition, rich in H (OH or H2O; e.g., serpentine, talc, chlorite, pargasite, phlogopite, brucite, phase A, phase B, phase D, phase E, phase 10A and so on). These DHMS can contain between 2 and 30 wt% H2O (average 10 wt%). NAM are mostly olivine polymorphs (wadsleyite and ringwoodite) or H2O-bearing garnets.
As concerns the total amount of H2O potentially stored, the NAMs are much more important because they are stable at "normal" geothermal conditions, while most of the DHMS are stable only at temperatures too low for a "normal" mantle at a given depth.