I don’t know why the animation does this, but the second character design seems pretty accurate to me, to how they are described when Frodo is wearing The One Ring. (For the Nazgûl themselves, the mounts are never winged horses)

When Frodo puts on the ring at Weathertop, the Nazgûl are described as having visible faces and hair, much like the second picture.

In their white faces burned keen and merciless eyes; under their mantles were long grey robes; upon their grey hairs were helms of silver; in their haggard hands were swords of steel.

The third was taller than the others: his hair was long and gleaming and on his helm was a crown. In one hand he held a long sword, and in the other a knife; both the knife and the hand that held it glowed with a pale light.

The Fellowship of the Ring, Chapter 11: A Knife in the Dark (emphasis added)

The second quote there is explicitly the Witch-King (aka The Lord of the Nazgûl).

But then the description at the battle of Pelennor Fields, when the Witch-King is seen in daylight by characters not wearing the ring, matches the first picture:

Upon it sat a shape, black-mantled, huge and threatening. A crown of steel he bore, but between rim and robe naught was there to see, save only a deadly gleam of eyes: the Lord of the Nazgûl.

The Return of the King, Chapter 6: The Battle of the Pelennor Fields

I don’t know why the animation does this, but the second character design seems pretty accurate to me, to how they are described when Frodo is wearing The One Ring. (For the Nazgûl themselves, the mounts are never winged horses)

When Frodo puts on the ring at Weathertop, the Nazgûl are described as having visible faces and hair, much like the second picture.

In their white faces burned keen and merciless eyes; under their mantles were long grey robes; upon their grey hairs were helms of silver; in their haggard hands were swords of steel.

The third was taller than the others: his hair was long and gleaming and on his helm was a crown. In one hand he held a long sword, and in the other a knife; both the knife and the hand that held it glowed with a pale light.

_{The Fellowship of the Ring, Chapter 11: A Knife in the Dark (emphasis added)}

The second quote there is explicitly the Witch-King (aka The Lord of the Nazgûl).

But then the description at the battle of Pelennor Fields, when the Witch-King is seen in daylight by characters not wearing the ring, matches the first picture:

Upon it sat a shape, black-mantled, huge and threatening. A crown of steel he bore, but between rim and robe naught was there to see, save only a deadly gleam of eyes: the Lord of the Nazgûl.

_{The Return of the King, Chapter 6: The Battle of the Pelennor Fields}