Awareness is exhibited in behavior. It's exactly due to the behavior be observe from LLMs that we don't ascribe them awareness. I agree that it's difficult to define, and it's also not binary, but it's behavior we'd like AI to have and which LLMs are quite lacking.

I'm assuming the "Wait" sentence is from the user. What I mean is that when humans say something, they also tend to have a view (maybe via the famous mirror neurons) of how this now sounds to the other person. They may catch themselves while speaking, changing course mid-sentence, or adding another sentence to soften or highlight something in the previous sentence, or maybe correcting or admitting some aspect after the fact. LLMs don't exhibit such an inner feedback loop, in which they reconsider the effect of the ouput they are in the process of generating.

You won't get an LLM outputting "wait, that's not right" halfway through their original output (unless you prompted them in a way that would trigger such a speech pattern), because no re-evaluation is taking place without further input.

Being aware and not admitting are two different things, though. When you confront an LLM with a limitation, it will generally admit having it. That doesn't mean that it exhibits any awareness of having the limitation in contexts where the limitation is glaringly relevant, without first having confronted it with it. This is in itself a limitation of LLMs: In contexts where it should be highly obvious, they don't take their limitations into account without specific prompting.