The second one is correct — the Earth is turning around, and the Sun only appears to move.
The merry-go-round of Activity 12.1 settles it. Sitting on a merry-go-round that turns anti-clockwise, you see every tree and building swing past you in the clockwise direction, entering your view on the left and leaving it on the right. You know perfectly well that the trees are not moving — you are.
Our situation on the Earth is the same, only slower and much larger. The Earth carries us around once in about 24 hours, from West to East, so the Sun enters our view in the East and leaves it in the West.
Why it happens: motion is always judged with respect to something. Since we cannot feel the Earth's smooth spin, we naturally credit the motion to the Sun. This is called apparent motion — the Moon and the stars show it too, and for exactly the same reason.
Did you know? Aryabhata explained this in the fifth century CE with a boat: a man moving forward in a boat sees the stationary bank slipping backwards. He even gave the length of one rotation as about 23 hours 56 minutes 4.1 seconds — remarkably close to the value accepted today.