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 you live in Italy anyways.
Imagine if a bank drew a boundary around its Manhattan skyscraper headquarters and declared itself a country called Bankistan whose only residents were janitors, financial analysts, and management-- and most of its citizens live in Brooklyn. Except for the C-suite and senior vice presidents who live in penthouses and the janitors who live in tiny rooms in the basement.
Also the second the bank fires you or you quit or retire, you're no longer a citizen of Bankistan.
At a minimum, a capital-see (heh) Country is something that belongs to you if but in a very, insignificantly, small part.
So my definition of "country" is ill-defined but does not include the Vatican.
Since countries are political divisions, I'd also argue that this is the main and most important definition.
BTW, even "continents" suffer from this where in the US, Europe and Asia they are defined differently.
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.