Which is correct: "Who is Greek president" or "Who is the Greek president"?

Solution 1:

There's actually a difference (or at least a distinction) between US President Barack Obama and The US President Barack Obama. In the former, the job title is being used as an attributive adjective phrase, similar to Six-foot-tall Barack Obama. The latter should strictly have a comma (The US President, Barack Obama) to show that the reference is to the holder of the office,and the name is just for identification. But it's very easy, particularly in journalism, to blur the distinction.

Solution 2:

Yes, it should. That is either a misprint, or written by somebody who is not a native English speaker.

The other example you give is different: it is referring to an individual, and there is the choice between treating "US President" as a modifier and treating it as part of the name.