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

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".

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.

First, your quoted text says:

"The actual term "Latin America" was coined in France under Napoleon III".

So Napolean didn't coin it himself, but it coined was during his reign.

In fact, another Wikipedia page gives a more detailed history.

1. In the 1830s it was postulated south America was inhabited by people of a "Latin race", and could become an ally with "Latin Europe" (Portugal, Spain, France, Italy) against "Teutonic Europe" (Germanic countries), "Anglo-Saxon America" (north America) and "Slavic Europe" (parts of central and eastern Europe).

2. In the mid- to late-nineteenth centuries, intellectuals and political leaders of south America used the term Latin America as they looked to France as a cultural model rather than Spain and France.

3. In 1856, a Chilean politician used the term in Paris and a Colombian writer used it in a poem.

4. The term was then supported by the French Empire of Napoleon III during the French invasion of Mexico to include France as a country of influence on the area and to exclude English-speaking countries, and to embiggen France's cultural and political role in the area.

In full:

Etymology and definitions

The idea that a part of the Americas has a linguistic affinity with the Romance cultures as a whole can be traced back to the 1830s, in the writing of the French Saint-Simonian Michel Chevalier, who postulated that this part of the Americas was inhabited by people of a "Latin race", and that it could, therefore, ally itself with "Latin Europe" in a struggle with "Teutonic Europe", "Anglo-Saxon America" and "Slavic Europe".[8] The idea was later taken up by Latin American intellectuals and political leaders of the mid- and late-nineteenth century, who no longer looked to Spain or Portugal as cultural models, but rather to France.[9] The term was first used in Paris in an 1856 conference by the Chilean politician Francisco Bilbao[10] and the same year by the Colombian writer José María Torres Caicedo in his poem "Two Americas.[11] The term Latin America was supported by the French Empire of Napoleon III during the French invasion of Mexico, as a way to include France among countries with influence in America and to exclude Anglophone countries, and played a role in his campaign to imply cultural kinship of the region with France, transform France into a cultural and political leader of the area, and install Maximilian of Habsburg as emperor of the Second Mexican Empire.[12] This term was also baptized in 1861 by French scholars in La revue des races Latines, a magazine dedicated to the Pan-Latinism movement.[13]

Now, none of this is specifically about English usage, so let's throw something in.

The earliest use I can find of Latin America in English is from 1863 in British and foreign state papers:

With the other States of Latin America, excepting the Empire of Brazil, which has a Legation accredited to this Government, we have no continued diplomatic relations, but we nevertheless have a lively interest in their welfare and prosperity.

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.