I'm not saying there are two, my point was that the legend differs from the truth. For all we know the 13th Lord Commander was killed by the Night King, who then took his place. If they want, they can spin it however they wish. If we ever see the Night King character in the books, he'll either be given another name (Great Other) or another origin.
At this point we just don't know enough, I think. If we saw the creation of the first White Walker, and that White Walker in turn became the Night King, then either the Night King and the Night's King are two different legendary figures or the two legends mingled somewhere down the line, either by chance of by purpose.
Yeah, I'm pretty sure that's what they're ultimately gonna go with. I don't see them spending show time explaining away that character, unless it's in the appendices again.