I'm looking up how the day duration works. So it seems I was right!
The day based on the rotation of the earth is called sidereal time (https://en.wikipedia.org/wiki/Sidereal_time) and is about 23 h 56 min long (I learned in passing that the rotation duration fluctuates every day, on the scale of milliseconds).
And the one based on the sun is the apparent solar time (https://en.wikipedia.org/wiki/Solar_time). It lasts up to 15min more or less depending on the season, but in average it's 24h.
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