What the Singularity Actually Means

The word singularity gets thrown around constantly right now. Most of the people using it would struggle to tell you what it means.
It gets used as a synonym for AGI. Also for ASI. Also for "the day AI gets really good," or "the day the robots take over," or just a vague wave at some future where things are weird.
The real meaning is more specific than any of those, and more interesting.
That's it. It's a claim about us.
Where the word comes from
Most people think Kurzweil. The Singularity Is Near came out in 2005 and sold a lot of copies.
He was about fifty years late.
The first use I know of is from 1958. Stanislaw Ulam wrote a tribute to John von Neumann after he died, and he was remembering their conversations:
One conversation centered on the ever accelerating progress of technology and…
Two years after the Dartmouth workshop that gave us the term "artificial intelligence."