President of [Country][Name] vs President [Name] of [Country]

It is unusual, but not to the point of being weird.

The most common usages are exactly as you say. It remains though that a title of Putin is "President Putin", and that he is "of Russia", and so it is a valid phrasing.

It's not done as a title, but as a clarification. They first refer to him as "President Putin", and then considering that there may be some readers who don't recollect just what he is president of, they add "of Russia".

Here is an example of a Whitehouse statement entitled "Readout of President Obama’s Call with Prime Minister Cameron of the United Kingdom, President Hollande of France, Prime Minister Letta of Italy, and Chancellor Merkel of Germany". In each case, it refers to each person by their title, and barring Obama (on the grounds that they assume anyone reading statements from the Whitehouse knows who the current US president is) they add their country.

Here is another that uses "Taoiseach Kenny of Ireland", despite no other country in the world using that title. To be (in English use at least) Taoiseach entails being Taoiseach of Ireland, but again some readers may not be familiar with the office, and so his country is given as a clarification.

Of course, people are likely to be more familiar with some countries than others, but that remains a reason to use it consistently; If one didn't use it of the countries best-known to your audience, but did use it of some others, there could be a perceived slight in suggesting that those for whom you added the country are less well-known because they are less important.