Who designates international airports?

It's more or less a functional definition: if there are flights between Airport X and some other airport in a foreign country, Airport X is an "International Airport". There are some "International Airports" that have the name largely by historical inertia, such as Kansas City International Airport (MCI), which was constructed to be a major hub for Trans=World Airlines before TWA decided to move its hub to St. Louis instead (there ARE international flights from MCI to Mexico, but only seasonally).

Most (but not necessarily all) international airports will have some form of customs and Immigration services, but this is not actually required.