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That’s a specific camera mode (action mode I think). Does the standard video mode also do heavy stabilisation?

Does it perhaps auto-enable when it deems it appropriate?

I don't have an iPhone, I've just noticed this (perhaps it's more obvious to me because I don't have one) in others' videos.

The stabilization is partially physical driven on phones this is called OIS. https://www.androidauthority.com/image-stabilization-1087083...

EIS is not usually needed for video but maybe in some cases it’s used?

I suspect iPhones do both. I have a 12 Pro, and when taking video the file is much more stable than the viewfinder while recording.