I was just doing random stuff on my phone and went to click a button near the top right of the screen, and was mildly horrified to see "1%", so I immediately put it into the charger, where the phone promptly started showing "0%" for the next ~30 seconds. The phone never died.
Has this happened to any of you?
Yup. Battery percentages are an abstraction over the actual voltage and remaining amp-hours - it's a complex formula which requires calibration and can easily be a bit off.
On one of my first smart phones, after replacing the battery with a much bigger one (both capacity and physically) it would still use the old formula, so it said 100% when halfway charged and after fully charging, during use it would stay near 0 (or at it - can't remember well) for half the actual usable time. I found an app that could show the actual voltage in the notifications, which helped a lot. (When it went under 3.5 or something, I knew it was almost out) But that number also varied with how much power was being drawn, etc.
Not entirely sure if this would apply in your case, but for future reference you can sort of "recalibrate" the battery percentage by:
I don't like doing that because it wears the battery. Better to have slightly inaccurate percentages than to actively make the long term situation worse
1 up-down cycle doesn't wear the battery, negligible. It would only count if you would do this every day. It's recommended to calibrate a new battery