What you have done here is cited the nonsense they were disproving in the episode. A transcription here shows the board was not patented as a spiritual or death communication board but as a novelty for amusement purposes See transcription of patent.
But it wasn't rebranded. It was referred to as the "Ouija or Egyptian luck-board" in the first of the two patents referred to above. The second one refers to "the ouija board, as is well known".
It was certainly never called the "Ouiji Board". That may or may not be a typo, but it was the name by which I often heard it referred to when I was growing up in the 1970s. In fact it was such a common error I'm surprised it hasn't turned up on one of those "Mandela effect" websites.