There are many ways of counting countries and some might use the 321 regions of the “travellers century club” or the 874 regions that “the most travelled” when they tell you how many countries they have been to. I think there is only one right way of counting countries, and that is by counting the 193 UN members plus the eight members of the UN specialized agencies (Niue, Cook Islands, Taiwan, Kosovo, Hong Kong, Macau, the Vatican and Palestine) which makes it a total of 201 countries in the World.
For example West Sahara is a big territory that is recognized as a country by many of the UN member states, but it is not a member of the UN or any of the UN specialized agencies, and does therefore not count as a country. The Vatican, although small is member of six of the UNs sixteen specialized agencies and is therefore a country. It is only the UN that can give a territory a status as a country and not other countries or organizations.
Anyway, these are the countries I have visited so far and in a few years I hope to have visited them all.