My best guess is that, when the Awoken were transformed, they saw the nature of both the Traveler and the Darkness – thereby being "awoken," or having the true nature of their struggle unveiled.
Exhibit A: the Awoken Queen has Fallen bodyguards. What if, in the Reef, there are species attributed to both the Traveler and the Darkness working together as a neutral faction? Perhaps not wishing to partake in the war between Light and Dark?
Exhibit B: We know that many Awoken returned to Earth to fight with the forces of the Traveler. Perhaps having seen the aforementioned "true nature" of the war, they chose to fight for the Traveler. If this was out of free will, then this theory would imply that there are Awoken who joined the Darkness, as well.