Nice that the new interval completely lies within the old.

I remember from college that there was an important physical constant where a new, more precise measurement was pretty far outside the range of the previous best measurement. I don't remember which one..

Depending on when you went to college, you may be thinking of the proton radius puzzle https://en.wikipedia.org/wiki/Proton_radius_puzzle which (to gloss over an enormous amount of detail) amounts to tension between two different ways of measuring the same property of the proton.

Other possibilities: Neutron lifetime, Newton's constant (twice!)

I think the story was that the first experiment was borked but done by someone famous, so the new experimenters didn't want to disagree too much.

This happened (allegedly) with the the Millikan oil drop experiment[0] which measured the charge of the electron.

[0] https://en.wikipedia.org/wiki/Oil_drop_experiment

This is a case when it's convenient to use the following notation:

    Accepted value: µ = 2.792847351(9)

    New value:      µ = 2.79284734462(82)

What's that notation called, and how does it work when digits overlap, e.g. in

    0.123±0.004

In your example one writes 0.123(4). I don't know what the notation is called, but it's discussed here: https://en.wikipedia.org/wiki/Uncertainty#Measurements

Makes sense, thanks.