Which country classification based on development is now obsolete and which country classification based on development is more widely used?
It is to be sure of the answer to this question because we do not know what classifications you have been exposed to. However, the most likely answer for the classification that is now obsolete is "Third World." This is a term that is no longer much used by social scientists because it is seen to have pejorative connotations, implying that somehow these countries are inferior to those in the First World.
Today, there are a number of terms that are more often used. Some people use the term "less-developed countries." Others use the term "developing countries." Still others use the term "South" to refer to those countries (typically, though not exclusively, to the south of the rich countries) that are not as rich and that have been exploited by the richer countries. Any of those three can be seen as the classification that is now widely used.