Have you ever played Pokemon like it's "meant" to be played?

No. I just believe the game canon (which there is none) and the anime and manga canon are completely different from each other.

The anime is different, it has nothing to do with the games storyline. It's the story of some other kid traveling the same places and possibly bumping into similar events.

The manga story line is an adaptation of the game's storyline. The names of the characters were picked from the original character designs made from the games. In other words, the names picked for the manga are the names the characters originally had when they were created. In the game, you're given the ability to name them whatever the hell you want. It also wouldn't make much sense to have the default name of the opposing game in the list of the game you purchased. So for the games, they put the first default name as the version you have.