I'm uncertain if Atari ST was the original platform, but BBC Micro, Acorn Electron, MS-DOS and Apple II definitely weren't as those ports were handled by third parties.
About Barbarian: The Ultimate Warrior
My guess is that the C64 version is the basis for all other versions and not the Atari ST version. Two reasons why I believe so: Firstly, the successor Barbarian 2 is, according to Wikipedia, a C64 original release with ports to other platforms and I don't see a reason why this shouldn't be the case with the even older Barbarian 1. Secondly, I played a couple of Barbarian versions on different platforms and the C64 feels absolutely as if this game was meant to be optimized for this platform, while other versions have this "I am a conversion"-feel".
What is quite certain is that the Atari ST is not the first one: the game was first released on CPC, C64 and Spectrum.