Except Criterion's print of "Anna Karenina" opens with a London Films logo, rather than the Fox logo, and slightly different credits. To my eyes, it doesn't look quite as clean as the Fox print (but it has been a while since I've watched the dvd). Criterion probably didn't have to license the title from Fox, since it must be part of the large London Films package they acquired. If they ever do put it out on dvd again, I hope it's the full-length version, rather than the cut one -- otherwise, what's the point.The version on Hulu is identical to the version on the Fox DVD.
BTW, if you have the Fox dvd, you may notice that some half-wits confused supporting player Sally Ann Howes with star Vivien Leigh. It's Howes' image, rather than Leigh's, that's all over the menu artwork. Sort of like shoving Thelma Ritter's face all over the "All About Eve" menus. (Though I admit Howes is a lot easier on the eyes than Thelma.)