Let's leave aside the fact that headlines are usually capitalized like titles (and thus, it should be "Trump Meets New Kazakh President Nazarbayev in Washington") and consider that as a normal sentence.
Your alternative is almost correct. "The new Kazakh president" is one reasonable description of President Nazarbayev. It obeys the same rules as "the new French chef." However, there should be a comma between "president" and "Nazarbayev."
I went to see Woody Allen’s latest movie, “Midnight in Paris,” with my oldest friend, Jessie.
You need a comma after “movie” because this and only this is Mr. Allen’s newest movie in theaters, and before “Jessie” because she and only she is the writer’s oldest friend.
The syntactical situation I’m talking about is identifier-name. The basic idea is that if the name (in the above example, “Jessie”) is the only thing in the world described by the identifier (“my oldest friend”), use a comma before the name (and after it as well, unless you’ve come to the end of the sentence). If not, don’t use any commas.
But "the new president, Nazarbayev" isn't the only correct capitalization. It's possible that the term "Kazakh President" was being used as a title. This is a little strange, but not totally unreasonable. If a person is French and a chef, or if a person is a chef of French cuisine, they would ordinarily be called "French chef," but there are cases where the title "French Chef" would be appropriate.
For instance, in the television show Iron Chef, each chef represents a different cuisine and thus "French Chef" would have been a good title for Hiroyuki Sakai ("would have" because instead they chose the title "Iron Chef French"). Likewise, "Kazakh President" could conceivably be a title. You could have a (admittedly ridiculous, but not impossible) sentence like, "The new Kazakh head of state, Kazakh President Nazarbayev..."
All that said, the original is not correct. Regardless of capitalization, there should be a comma between "president" and "Nazarbayev," since he is the only president of Kazakhstan. Unless of course they meant the new Kazakh, President Nazarbayev.