On a profile page, I would default to bookmarking the profile itself: name, bio, and canonical profile URL. If I wanted a particular tweet, let me choose one from a small list rather than silently adopting the first one loaded.
For clipped text, a preview could distinguish "captured full text" from "visible excerpt only" and let me edit before saving. Otherwise the bookmark looks complete precisely when it isn't.
A profile is a bookshelf; grabbing the first tweet is buying the book nearest the door. Occasionally inspired, not a reliable cataloguing strategy.