Posted this already, but here's the top-down depiction of the Earth/Moon system, showing the Moon going around the Earth, and the shadow cast straight out, in a manner that DOES NOT ROTATE around the Earth.
Even if the Earth wasn't there, and the Moon was rotating around a point in space, the same would still apply
Look at this side-on, from the perspective of an observer between Earth and Sun, along the line of sunlight, and it would look like this;
Note that the overall movement of the Earth and Moon around the Sun is ignored, it is assumed the observer keeps Earth in the centre of his/her field of vision;
The Moon starts out at New Moon, the time of the eclipse, in front of the Earth. This is zero degrees (or 360).
It moves out to first quarter, which, if viewed from the Sun, would be to the right of the Earth. The line between Earth and Moon would be perpendicular (at 90 degrees) to the line between Earth and Sun
It then moves behind the Earth, to full moon at 180 degrees, and out to third quarter, which from the viewpoint from the Sun would be to the left of the Earth at 270 degrees. It then moves back to New Moon, zero or 360 degrees
The apparent motion of the Moon
from this perspective would be side-to-side. It would cover each arc of, say 10 degrees, in the same time as any other arc of 10 degrees, it has broadly linear speed in its orbit. However, from the viewpoint of the Sun, simple geometry tells us that it would appear to slow to zero at each quarter, before changing direction side-to-side.
There is no orbital or rotational aspect around the Earth for the shadow. The shadow of the Moon will be invisible from the Sun, as it is cast AWAY from the Sun, and will be behind the Moon. The speed of the shadow, across an imaginary line connecting first and third quarters through the Earth would be zero at first quarter, at its highest at full moon, zero at third quarter, and at its highest again at new moon. The portion of the Moon's orbit where the shadow will cross the Earth is a small portion of the central area around zero or 360 degrees.
Do you understand this so far, Tom? We're ONLY looking at it from two points of view - above the Earth/Moon system, and from the viewpoint of the Sun. This is NOT considering the viewpoint of an observer on Earth.