The Earth does not turn through a minute of arc or an arcminute during a minute of time, nor does it turn through a second of arc or an arcsecond during a second of time.
Assuming that ther is a connection between arcminutes and minutes of time is an error.
In angular measurement:
A circle is divided into 360 arc degrees, each arc degree being 1/360, or 0.002777, of a circle.
Each arc degree is divided into 60 arcminutes, each arc minute being 1/(360-X 60) or 1/21,600 or 0.000046296, of a circle.
Each arcminute is divided into 60 arcseconds, each arcsecond being 1/(360 X 60 X 60), or 1/1,296,000, or 0.000000771, of a circle.
In time measurement:
One day is divided into 24 hours, each being 1/24, or 0.0416666, of a day.
One hour is divided into 60 minutes of time, each minute of time being 1/(24 X 60), or 1/1,440, or 0.000694444, of a day.
One minute of time is divided into 60 seconds of time, each second of time being 1/(24 X 60 X 60), or 1/96,400, or 0.000011574, of a day.
The Earth turns 360 degrees of arc in one 24 hour day of time.
So the Earth turns through 15 degrees of arc in one hour of time.
And thus the Earth turns through 15 arcminutes during one minute of time.
And so the Earth turns through 15 arcseconds in one second of time.
So any source implying that an arcminute is equal to the angle that Earth turns in a minute of time is incorrect.
The Earth does not turn through one arcminute in one minute of time, nor through one arcsecond in one second of time. So a different number of arcminutes fits into a full circle than the number of minutes of time during a day.