Your definition ignores international recognition.

Ultimately, whether you get to act like a country (go to the UN, engage in diplomacy, hold territory) is in large part based on whether other countries recognize you as such. I don't know that it defines country-hood but it's part of the puzzle.

The Vatican is a fascinating example since it's clearly a very different sort of entity than the rest of the countries, yet is still recognized by most of the world's nations.

Historically the Vatican is also a way to allow the Papal States to “give up” their claim on, well, the Papal States.

It works out pretty well for everyone, so it continues.