How important is story in a video game for you?

thats a fact jack!

I find it too distracting to interject a story into my interactions.

and that is aside from bad story telling and how hard it is to do in a game where a player has freedom in the first place

There are arcade games with stories, and non-arcade games without stories. Story is optional.

Pretty much this. It depends on genre. For visual-novel and graphic-adventure games, story matters more. But for almost every other genre, gameplay matters most.

Here's the thing: Dark Souls has an incredible ability to tell its story through gameplay. It not only has consistently good gameplay, but is able to tell stories through its mechanics and through going through its environments. Dark Souls probably wouldn't have been as popular without its incredible atmosphere and subtle storytelling.

Story does not bother much for mp play a la Ut, otherwise it's an important factor for me...

It's the most important factor in my opinion. Its why I prefer The Last of Us over virtually any game out there. Or how I liked MGS4 more than MGSV: The Phantom Pain, the latter has vastly superior gameplay but no story...

Same goes with games like FFX, Bioshock, Bioshock: Infinite, older Metal Gear games etc. I'll never forget a good story.