Yes. If he talks with Putin about "government policy" or "international relations" then he has to register with the US government as doing so.
It's against the law to privately speak with foreign nations about those categories. It may put you at odds with US policy or maybe you dont see the whole picture, so it's better in the US eyes to ban foreign diplomacy by private citizens.
It reminds me of that old joke of the 2 scientists studying a fly.
Scientist 1: Fly, fly! Scientist 2: When fly has two wings fly flies 2 feet.
Scientist 1: pulls off one wing and says Fly, Fly! Scientist 2: When fly has 1 wing, fly flies 1 foot
Scientist 1: pulls off the other wing and says, Fly, fly!. Nothing happens Scientist 1 Fly, Fly! Nothing happens Scientist 1 showing frustration> FLY! FLY! Scientist 2: When fly has no wings, fly becomes deaf.
Bad-dum!