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Technically, they all take place on Earth in the same universe within the Virgo supercluster of galaxies... That said, the games aren't really connected story/campaign-wise, except for the 3 original Modern Warfare games.
However, your points don't carry over from one version to another. The most important advantage on the Battle side is the supposedly superior performance in default settings. Players claim that 2024's MW2 ran better on Battle than it did on Steam.