Re World Series
The name "World Series" is a marketing creation designed to get people to sit in a cold stadium on chilly October nights. The obvious names for it, "American" and "National" were already taken by the leagues. "Federal" might be an option, but you can see why the marketing and hype men went with "World".
At the time (we're talking 1903 and earlier) baseball was only in America, so by the same logic that we have a Miss Universe contest, the winner really was the best baseball team in the world. Even today, the claim is still valid. The two Japanese leagues are not at the major league standard, and nothing else comes close.
The logic doesn't carry over to football, where the Premiership is just another national championship on a par with La Liga or the Austrian League or Japan's top flight. Even the UEFA Champions League winners can only claim to be Champions of Europe, since that competition is on equal standing with the same competitions around the world.
That is simply not the case in Baseball, which doesn't change the fact "World Series" is nothing more than a marketing name anyway.