ZadokAllen: Definitely the original. The extra bits in DC are fine, but I personally found the shifts in art-style and the audio quality being all over the place to be very distracting.
Plus... bit of a spoiler here... by the half way point they forget about the extra content and you just play the original, with maybe one or two added screens, until the end anyway.
I assume that it would have been much more difficult to include additional events of the extended storyline once the story reached certain point and besides at that point they already had told that additional story. While it wasn't much, it was certainly more than what BS2: Remastered version included (comic which explains what happened before the events of the game).
However weakest thing in Director's Cut versions is that they leave out stuff. BS1: DC removes some dialogue from original game and Remastered version of BS2 removes one maze puzzle. And that's the reason why I recommend playing the originals first.