Basic D&D has an odd history.
The original (Eric Holmes) edition was designed as a simple version of the original D&D game; it was designed at the same time that Gary was designing AD&D, so there are some elements of AD&D in it. It was really intended to point people towards AD&D. It had no world attached to it.
The next edition (Tom Moldvay) divorced it from AD&D, and really made it a separate game, drawing inspiration from the original D&D but doing wonders for organisation and simplicity.
David "Zeb" Cook designed the "Expert" set, and Cook and Moldvay wrote the original module for that setting - X1: The Isle of Dread.
The third edition of Basic D&D (Michael Mentzer) is the classic "Red Box" edition, though I feel the Moldvay edition is superior in writing style and presentation.
Now, somewhere around now, a map of the "Known World" appeared. Certainly one appears in my (Mentzer) edition of the Expert set and the Isle of Dread - I don't know if one appears in the Cook edition of the Expert set.
With the release of the Expert series of modules, the Known World began to grow. It wasn't a world that came from an existing campaign, but rather driven by the map provided in the Expert Set/Isle of Dread. Eventually the world got called "Mystara" - I believe as it was converted to AD&D 2nd edition and they realised it needed a name.
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AD&D 1st Edition derives mainly from Gygax's personal campaign - which happened to be Greyhawk. Of course, Gygax's Greyhawk is substantially different from the published Greyhawk!
AD&D 1st edition does transcend Greyhawk, though. A classic example would be the Desert of Desolation modules by Tracy Hickman (with others) - that's about as non-Greyhawk as you get, whilst remaining in the 1E mode.
I think that there are many different styles of 1E module, and thus the feeling you get from 1E can be completely different depending on the module. Compare Dragonlance to the Bloodstone Wars to White Plume Mountain to the Sinister Secret of Saltmarsh.
I think 3E supports all those styles, however, the Wizard writers of modules aren't really doing the system a great service with the modules they've created. However, part of that may simply be nostalgia!
Cheers!