There is nothing to indicate that the White Walker shown being created, was created before the Wall was created. Or that he was a Lord Commander of the Night's Watch. Or that he's actually the Night King.
The show may diverge from the books in this, as the show has been known to merge or separate characters as needed. (Examples: The Sand Snakes, Jaqen and the Kindly Man)
What we do know, mostly coming from the Blu-Ray special features (The Children of the Forest, the First Men, and the Andals and The History of the Night's Watch).
First came the Children of the Forest, second came the First Men from Essos. War happened as the First Men chopped down the trees of the Children 12000 years ago. War lasted for 2000 years, then a peace for 2000 years. Then the Long Night came:
"Thousands of years ago, there came a night that lasted a generation. Kings froze to death in their castles, same as the shepherds in their huts; and women smothered their babies rather than see them starve, and wept, and felt the tears freeze on their cheeks... In that darkness the White Walkers came for the first time. They swept through cities and kingdoms, riding their dead horses, hunting with their packs of pale spiders big as hounds."
-s01e03 "Lord Snow"
The Long Night resulted in the White Walkers coming to attack. This was followed by the "The War for the Dawn":
None knew why the White Walkers came, nor their nature and origins, but they killed all in their path. The White Walkers reanimated the dead as Wights to kill the living at their command, and soon the White Walkers led their hordes of undead to sweep across the continent.
The First Men drove back the White Walkers to the furthest north
Eventually the First Men and the Children of the Forest rallied to defend themselves and in a conflict known as the War for the Dawn, the White Walkers were defeated and driven back into the uttermost north.
The Wall was constructed along the northernmost isthmus of northern Westeros to bar their return, a massive fortification standing seven hundred feet high and stretching from one side of the continent to the other. Legend says that the Wall was infused with powerful magic spells by the Children of the Forest that prevent the White Walkers from crossing it. The ancient order of the Night's Watch was founded to defend the Wall should the White Walkers return to invade the realms of men once more.
So at 8000 years, the Children and the First Men teamed up. These First Men split into what we know as the Northerners (The Starks have heavy First Men blood in them) and the Free Folk/Wildlings.
Then at 6000 years ago, the Andals came (The Lannisters have heavy Andal blood in them). And the Andals began to kill the Children again, who were already experiencing a low population after the first war with the white walkers.
Given both of these events, the Children likely created multiple waves of White Walkers. The first before the Wall, to attack the First Men, and then again afterwards, to attack the Andals.
In the latest episode, we see Leaf creating a White Walker, possibly the first time round. Nothing says he becomes the Night King, or that he was a Crow.