Why did Napoleon III coin the term "Latin America"?

The actual term "Latin America" was coined in France under Napoleon III and played a role in his campaign to imply cultural kinship with France, transform France into a cultural and political leader of the area and install Maximilian as emperor of Mexico. — Wikipedia

Solution 1:

Latin means "a native or inhabitant of a country whose language developed from Latin." It includes also inhabitant of countries like France, Italy, Spain.
Hispanic means "of or relating to Spain or to Spanish-speaking countries." It can be used to refer to Spain, but not Italy or France.

Naming those lands "Latin America" was a reason for Napoleon III to claim possession of the lands.

Solution 2:

The reason is that Latin America was formerly known as "Hispanic America", thereby implicitly excluding France.

By "rebranding" it to "Latin America", France would be allowed to claim some paternity or at least influence, this in turn would justify the Mexican adventure.

I'm not sure how a Habsburg such as Maximilian would qualify as "Latin".

Solution 3:

Everyone is over thinking a rather simple answer. The term Latin America was coined to denote that the countries in the region all speak a "romance language" derived from Latin. This includes Spanish, Portuguese, and French. Nothing more, nothing less.