I am a first year PhD student and I was recently working on reproducing some results from a paper which is still being reviewed. In the process I noted a number of issues with the paper which should be clarified or fixed. I have been asked how I would like my name to appear in the acknowledgements. My full legal name is Kristian Tyn-Kai Chung, but I do not ever use the name Kristian, I have always gone by Kai to everyone I know besides by mother and grandmother who call me Kristian (with a Norwegian pronounciation). Should I use my full name? First and last? Should I use Kai? Any suggestions appreciated.
If you have already published papers, then use the same name as on your papers.
If you haven't, then now is a good time to decide what you want your "professional name" to be. There's no particular need for it to match your legal name. You might just think about what you want people to call you in academic or professional situations. (And if it's a common name, you may want to choose a more unique variant; e.g. including a middle name or initial.)
But if possible, choose something that you think you'll be comfortable using for the rest of your career. It's awkward to change your professional name, because you can't go back and change your name on papers you've already published, and people will have a hard time connecting the work you've done under the two names.
(This matters less for an acknowledgment, though, since it's not really that important for people to know that it's the same person as on your later papers. So you don't actually have to worry about it too much right now. But it's a decision you'll have to make eventually; might as well do it now if you're ready.)