Yes you can play Game Cube games on the Wii. The reason you can do this is because the Wii is backwards compatible with the Game Cube. You can not however play Wii games in the Game Cube.
There are two different Wii's right now. The first one is the original one capable of playing Wii and Gamecube games. This is RVL-001.
It's the Wii that comes with the stand so it can sit vertically. RVL-101 is a revision that removes all gamecube related features. Which means you can't play Gamecube games or use gamecube accessories along with whatever is on that support page that's stated it can't do.
The current one Nintendo has on the Wii section of the site is the revision. They've stopped producing the old one so you can only get this one unless you get it used or somewhere where they still have it in stock.
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