There would still be a lag with the stationary system, but the apparent and actual position would be identical.
What if you made the same observation but from the opposite side? [sunset position]
As before, the apparent position and actual position would be identical.
What if you stood on the side nearest the Sun so that the Sun was overhead [noon position], what about then?
Same as before, the only thing that changes is the direction that the observer is facing.
If that is not the case, is the lag due to angular rotation speed?
The lag is due to the time it takes for light to travel from the Sun to the Earth combined with the Earth's angular rotation.
Or is it due to the surface speed of the observer due to rotation?
Just the surface speed of the observer combined with the velocity of the Earths orbital speed gives you that tiny ~.006º difference, that is not what I am trying to explain.
“The lag is due to the time it takes for light to travel from the Sun to the Earth combined with the Earth's angular rotation.” [excerpt]Is the lag present when observed from the North or South poles?