I'm so confused about this. I googled it and read about knight's tours, however they all start from illegitimate positions. I want to know if a knight can move through all squares from its original position (e.g. b8, g8, b1, and g1).
Yes, it can
This particular knight's tour is closed, meaning that it starts and finishes in the same square. Therefore, the knight can start at any square on the board and finish on the same square, since it just starts at a different point along the cycle.