I still think the original definition is the most useful, and also gives the greatest intuition as to
why it is called the "singularity" and not some other name.
"It is a point where our old models must be discarded and a new reality rules, a point that will loom vaster and vaster over human affairs until the notion becomes a commonplace." - https://frc.ri.cmu.edu/~hpm/book98/com.ch1/vinge.singularity...
What I like about this definition is that its philosophical utility is precisely that is a name for the place where our expectations about the future break down completely, and this is a useful concept to have a name for.
It is true that the most commonly explored particular manifestation of this is an AI technological breakthrough, but using the definition contextualizes that as just one possibility in a set of possibilities, including some "Singularities" that have already happened.
In particular, try to imagine explaining the modern world to a Sumerian peasant farmer in X,000 BC. Good luck with that. You probably don't understand the world they live in very well either, but they can't even conceptualize our world. There's at least one Singularity in human history between that farmer and us, and it's not hard to draw out at least a few more.
However, it can be challenging to point out "where" or "when" they occurred.
From this we can draw out a few more lessons. Singularities are relative to the observer. You generally can't "pass through" one, because as you would approach one, your ability to prognosticate the future reasonably grows through it, and so relative to you, in time, it recedes. Even if the hard AI takeoff is going to happen, we are even now learning more about what an AI world looks like and how it works, so even in that scenario, our ability to understand it improves as we approach it. Hard, species-wide singularities that give no warning and no opportunities to significantly adapt in advance have generally not happened yet, even as like I said, clearly, between that Sumerian farmer and us there has to have been at least one.
Hard AI takeoff is one. But another one that is back in the news is nuclear war. That would also constitute a singularity in its own way; we could fall back on our understanding of previous civilizations at lower tech levels but it's still very difficult to predict what would happen in the aftermath of such a thing. There's a few other possibilities. But what one might call a "hard" singularity, one that completely blindsides the entire species, hasn't generally happened, and there's still good reason to suspect they're not likely.
As this post demonstrates, I find this formulation far more interesting for thinking and saying things than the vague "future tech might be weird" ideas that tend to float about. You need the concept of anchoring the singularity to particular points of view to be able to say much about them, I think. Trying to treat it as an objective event is difficult, precisely because they recede from the observer.
This formulation also disposes of the "Rapture of the Nerds" scenario. There is no reason to believe that the singularity somehow must be accompanied by "transcending the physical world", there is no reason to believe that such an outcome would even be good for humans in specific or humans in general, and there's no reason that singularities must be limited to technological process. That's just one particular part of the space (and IMHO, a very small one when it comes to hypothetical transcending the physical world) that happens to have some scifi stories written about it, but is far from the most likely possibility.