'Anchor' is indeed the most up-to-date term, now in the process of replacing the sexist and already slightly dated 'anchorman.' ('Anchorwoman' exists, but it's comparatively long, awkward, and therefore not all that common, much like 'chairwoman,' which has also never caught on strongly.)
The anchor at an American TV network is the journalist who sits behind a desk and leads the broadcast of a national news program, not only reading the main news but writing it and usually also having considerable, though by no means absolute, editorial control over content (choice of topics) as well. Famous past anchors include Walter Cronkite and David Brinkley, now both elderly; more recent are Dan Rather and Tom Brokaw (both recently retired), Peter Jennings (died not long ago), and Jim Lehrer (close to the older generation but still on the job at PBS). The current trend is away from a single 'star' anchor and towards a pair of anchors or a rotating team.
Local stations also have anchors who lead the local news broadcast; these are sometimes chosen more for their photogenic qualities than for their journalism experience, so some of them may have correspondingly less editorial control, but then again, that trend is fast advancing on network and, especially, cable TV as well.
Whether this is more like a Nachrichtensprecher or a Moderator is for German speakers to say. If, as I suspect, it's somewhat like both, then why not include both options? Very few words have a single 1:1 translation.
'News reader / newsreader' in this sense, though, should definitely be marked BE if it isn't already; the main sense of it in AE, as far as I'm aware, is a software program that downloads and sorts e-mails from online newsgroups such as Usenet.