Historically, a day was defined as 1 earth rotation.
And that's where you went wrong. Historically, a day was one sunrise to the next.
That doesn't make sense since humanity have been living on parts of the Earth with constant night during winter and constant day during summer since god knows when. "One sunrise to the next" doesn't make sense there. There were probably local units all over the place.
Read this very carefully as you quoted only part of it. "Between 1000 (when al-Biruni used seconds) and 1960 the second was defined as 1/86,400 of a mean solar day". So, that means before the year 1000 the second did not exist as a defined standard; and may not have existed at all; still fuzzy on that.
I have a hard time imagining all people on Earth of the 11th century all agreeing on the definition of time units at all, let alone on the second. It's probably one of those things where the now standardized time units of hours, minute and second was used more and more, and gradually took over any existing local units and standards.
But why 60ths? Was a minute always that long?
I imagine it's because it was convenient to divide the minute in the same way as the hour. There's 60 minutes in an hour thanks to the Babylonians, who used that as their base (just as we use base-10).