My current understanding is this: if I know everything I need to know to achieve some goal, then I am situation-aware.
That is correct. Situation Awareness (SA) is described quite well in Wikipedia.
Situational awareness or situation awareness (SA) is the perception of environmental elements and events with respect to time or space, the comprehension of their meaning, and the projection of their future status.
Situation awareness has been recognized as a critical, yet often elusive, foundation for successful decision-making across a broad range of situations, including aviation, air traffic control, ship navigation, health care, emergency response, military command and control operations, and offshore oil and nuclear power plant management. Lacking or inadequate situation awareness has been identified as one of the primary factors in accidents attributed to human error.
Basically, if you have situation awareness, you are aware of every single possible aspect involved in a situation and therefore you have a much better chance of predicting the future outcome of a given method of moving forward.
You also asked if you would need SA for everything you do and the answer is no, but ideally you would need at least some SA.
As Wikipedia points out in the link and quote above,
Lacking or inadequate situation awareness has been identified as one of the primary factors in accidents attributed to human error.
For an example, if you were to cut yourself deeply, and you were bleeding out, without any SA you would more than likely die. With some SA, you would know that you have a better chance of survival if you try to stem the flow of blood somehow, but how?
You may think that just putting your hand over it is enough, but if it is an arterial bleed, would it really be enough? If emergency medical assistance comes quickly enough, then maybe, but maybe not.
A tourniquet would stop the flow completely, but that can also cause issues but do you know that and what can you do to mitigate those issues (if at all).
Knowing all the possible methods and pitfalls of carrying out a task will provide the very best possible outcome; but if you really need to make a decision right there and then, you can always make an educated guess if you have some idea of how things can go.