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We have seen the first white walker, created by the the children of the forest.
We also see a white walker who was supposed to be the Night King several times.
Based on their faces, can they be the same person? I know the best canon explanation for the Night King is him being the 13th Lord Commander of the Night's Watch, however, this can easily be a red herring.
Any canon evidence, supporting/disproving the first white walker being the Night King is also welcome. Although the images are from the series, book explanations are also welcome.
EDIT: Actually my question is very different and focused compared to the differences between books and show. It is already somewhat answered, both characters played by the same person.