I thought
Katherine was more of the original (though I suppose the "K" could still be "modern-looking").
I think it doesn't matter if you go by your full name. I also think that for some short forms, you don't have to have Katherine/Catherine as a full name.
Kate shall suffice. However, if you're going to go by a short form with Katherine/Catherine as your full name, the first letter should probably match (so
Catherine would be
Cate, not
Kate).