A Time-aware faction or civilization is one with some knowledge, if not influence over time-active powers, which could, for example understand the significance of the Last Great Time War even if they had no part to play in it.
Aside from certain select organizations, Humans did not even partially grasp the significance of the various temporal factions for much of their history. By approximately 2.000,000, Humans had some awareness of Time Lords, though representatives of that species, such as Sabalom Glitz tended to think of them as almost magical, at the same time not seeming to take them entirely seriousy. (DW: The Mysterious Planet)
By contrast, five billion years later, Jabe, a member of another, Earth-derived species, the Forest of Cheem, on the contrary seemed cognizant of the Time War and the loss created by it, in apparent contrast to the Humans of that era. (DW: The End of the World)