Earlier quoted context omitted.
What is a country for you? How do you define it? Is the land size that important?
A country must have a people. Who are the people of the Vatican? The only persons who live there are temporary government employees and not even all of them are citizens because that is optional. You cannot own property, vote for your government, start your own business, go to school, buy anything except what is stocked in the small canteen, or go to the hospital if you are a Vatican citizen and odds are pretty good…
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.