It’s a difficult question to answer. Is Belize a developing country? Definitely. But is it a Third World country? Probably not.
Originally, the Third World was used to denote the countries that weren’t aligned with either the USA or the Soviet Union the Cold War. This definition no longer applies and is fallen out of use. Under this definition, Belize is undoubtedly a Third World country.
The term “Third World” has largely been replaced by “developing countries” and “least developed countries.” Although one may consider developing country and Third World to be synonymous, and the description of developing country sounds quite damning (low levels of access to safe drinking water, sanitation, and hygiene; energy poverty; high levels of pollution, etc. etc.), China and Russia are considered developing countries. So considering this definition, Belize would be considered a developing country.
If one tries to categorize the Third World by the least developed countries, Belize does not fall under this category. Other sources categorize the Third World by gross national income, human development, and press freedom. Based on their assessments, Belize does not fit in these lists either.
In short, Third World is somewhat outdated and not a very useful term, as well as being hard to define; and unless you’re using quite a broad definition, Belize is not a Third World country.