Well it's complicated Galen but can I simply the issue thus:
The simplest systematic name of water is hydrogen oxide as you mentioned. This is analogous to related compounds such as hydrogen peroxide, hydrogen sulfide, and deuterium oxide (heavy water). Another systematic name, oxidane, is accepted by IUPAC as a parent name for the systematic naming of oxygen-based substituent groups, although even these commonly have other recommended names. For example, the name hydroxyl is recommended over oxidanyl for the –OH group. The name oxane is explicitly mentioned by the IUPAC as being unsuitable for this purpose, since it is already the name of a cyclic ether also known as tetrahydropyran.
The polarized form of the water molecule, H+OH−, is also called hydron hydroxide by IUPAC nomenclature.[47]
Dihydrogen monoxide (DHMO) is a rarely used name of water. This term has been used in various hoaxes that call for this "lethal chemical" to be banned, such as in the dihydrogen monoxide hoax. Other systematic names for water include hydroxic acid, hydroxylic acid, and hydrogen hydroxide. Both acid and alkali names exist for water because it is amphoteric (able to react both as an acid or an alkali).
Has that answered your question?