I live in a country where most people are either atheist or christian. Estimates are that between 0.4 and 1% of all people here call themselves a Buddhist.
It's been my experience that there isn't a single agreed upon definition about when you can call yourself a Buddhist. Some people say you are a Buddhist if you consider yourself to be one, others say you need at least several years training from an acknowledged Buddhist teacher. This being said, most people I know (but certainly not all) regard the ceremony of 'taking refuge' as the moment when they became a true Buddhist.
Personally I like the view of Dzongsar Jamyang Khyentse Rinpoche who has written a book on the topic called "What Makes You Not a Buddhist":
It’s whether you agree with the four fundamental discoveries the Buddha made under the Bodhi tree, and if you do, you can call yourself a Buddhist (source).
Disclaimer: this answer is largely based on my own answer on a similar question