A "train of thought" can be seen as a trace of a depth first search where the preceding trace is used to guide termination and next expansion decisions. A similar concept, "taboo search", exists in classical constraint optimization where previous solutions are fit to a model that guides future expansion (but as the name "taboo" implies, away from uninteresting solutions).
We also have harnesses that perform breath first search.
If I tried to describe what it means to "think deeply", I would probably say a combination of both.
Ultimately I believe that we will surpass human capabilities but fail with alignment. Handing the world's resources over to stochastic systems that can evolve faster than we can reason about them simply leaves too many "interesting" outcomes that do not end well. I also expect the failure modes will be totally non-obvious.
The other option is that you do understand those words the same way, and the people making these (now nonsensical) anti-AI claims simply aren’t talking about the same programs/models we are. Their idea of SOTA is when chatgpt.com launched.
If you took a point sample pre-Opus, and didn’t write a good prompt, of course you would think all AI programming was worthless slop.