Tarot cards being reserved for fortune telling is a relatively modern and very American concept. They began as, and in Europe still are, playing cards. https://en.wikipedia.org/wiki/Tarot_card_games
I realize it can go in either category, so there's not really anything to resolve here, but it much be noted saying it's not a game is a bit silly.
The very page you used mentions a French coming with the concept and mentions occultists in general, a belief spread around the world, believes in the power of tarot. Also, I'm very familiar with how tarot is very spread on Japanese pop-culture, though I don't doubt it may be american influence. Nonethless, I do not think it will alter the position where we put The Magician, though it would be easier to say it's a motif used through media.