So the theory explained here is that Hydrogen mixed with the Oxygen in the melted rock (magma) of earth, under extremely high pressure to create our earth specific flavor of H2O (appropriate amount of trace minerals and deuterium).

Am I reading that correctly?

Link to the paper mentioned in the article: https://www.nature.com/articles/s41586-025-09630-7

Correct, and we can demonstrate this via various gem-bearing and REE-bearing pegmatites which almost universally contain magmatic-sourced water trapped within them.