It is a very interesting problem that, IMHO, most militaries go about in the wrong way as a result of a) how the military PME is structured and b) the assumed requirement to have a top down model. As a bit of context for my comments, I've taught cultural Anthropology on and off for 15 years and, from what I have seen of military PME in the area, it is pretty poor on the whole.

I suspect that part of the poorness of most cultural training (awareness, understanding, competence, etc.) is due to a really poor understanding by many people of what they actually mean by the terms. Let's take those three terms or levels that you mentioned, and I'll give you my take on how I think they should be understood and what the consequences of such an understanding can be (something usually not talked about).

Cultural Awareness

At its simplest, all this refers to is that the student / soldier should realize, in their gut as well as their head (thumos vs. logos if you want to go all Greek ), that people can and do organize themselves in different ways to achieve similar goals (actually, homeostatic, psuedo-end states).

As a consequence of this realization, students will tend to start questioning whether or not the way their own culture / society organizes to meet a given need is the "best" possible way to do so. Generally, people will still stay in their "comfort zones" (i.e. their own cultural responses), but they may get attracted to some of the "fringes" (low frequency distribution options).

Cultural Understanding

Again, in its simplest form, this should refer to two things. First is a mental set of rules or patterns on how cultures organize to meet basic needs; something along the lines of a broad topology of cultural organization, with some basic rules for how to analyze the various forms of cultural organization. The second thing is more of a "mental" or "psychological" ability of the student to move into the "head space" of at least part of another culture; think verstehen as used by Wilhelm Dilthey). This goes back to Dilthey's observation that"

All science is experiential; but all experience must be related back to and derives its validity from the conditions and context of consciousness in which it arises.
Source
So, using this definition, someone who has achieved "cultural understanding" should be able to a) make sense of a given culture and b) be able to mentally "shift" into it after some time spent immersed in that culture.

The primary consequence of this type of understanding is that people are more likely to "detatch" themselves from their own cultures.

Cultural Competence

To my mind, "cultural competence" is a more generalized development of "cultural understanding"; sort of a graduate degree version of an undergraduate "understanding". Someone who has achieved this, however you measure it, tends to be abstracted from their own culture - they can act within it, but it does not have the "aura of Truth" that a culture needs in order to control its members. As an example, Anthropologists who have done a fair bit of fieldwork tend to tal;k of themselves as standing "betwixt and between" cultures; people who move between them, but are not solid, absolute members of any of them even while having strong ties to multiple, often conflicting, cultures.

At any rate, that's my take on how these terms should be used; probably more of a hindrance for you than a help but, then again, I'm an Anthropologist .

Cheers,

Marc