>>> I'm not anthropomorphizing anything ...

Yes you are, regarding LLMs at least. Here's why:

  just for fun I asked a reasonably smart LLM to ...
  [be] capable of understanding if it's gone off on
  a hallucinatory path ...

You then went on to state:

> If you know something rare and the LLM does not, you'll immediately see when it's hallucinating an answer or answering factually.

Again, "hallucinating" is not something an algorithm can do. Also, determining factuality is again subjective based on the person assessing the information.

You ever heard of something called a metaphor, guy? I use the word "smart" as shorthand to describe something that scores highly in a number of coding and terminal use benchmarks (as compared to, let's say, a 30B size model from one and a half years ago which will score much worse), and "hallucinating" to mean "outputs plausible sounding gibberish that doesn't hold together consistently". Of course there's no actual hallucination going on.