The following explanation was given to me by many Latin Americans: Before Columbus discovered the Americas, the races were geographically distributed in discrete groups. The "whites" in Europe/Middle East, the "reds" in the Americas, the "blacks" in Africa, the "yellows" in the Far East, etc.
After October 12, 1492, all the races melted together in the Americas. Therefore, Columbus Day is known in Spanish as the Day of la Raza , the day when the human race became melted again.
Therefore, la Raza looks different depending on the part of the Americans being studied. In Argentina, la Raza looks like a European mix, while in Dominican Republican it looks like an African/European mix. In Mexico and Central America, la Raza mostly looks like an Indian/European mix.
As you stated, la Raza can mean different things to different people. In California and the South West, it seems that la Raza mostly refers to the Mexican-Americans. In New York, Chicago, and Orlando, la Raza must include the large colonies of Americans of Puerto Rican origin. In South Florida, la Raza must be the a complete mixture of all the Latin American countries, since South Florida has a large contingent of immigrants from every Spanish-country in the world, from very pale blonds to very dark blacks.
The origins of "La Raza Cosmica" were created by the Mexican intellectual Jose Vasconcelos. It was used to unite the people during the revolution. It was pure government propaganda and unique to Mexico.
I can't believe people actually take this organization seriously. They get all of their funds from the Ford Foundation and other corporate foundations and have as much popular support as a bottle of whiskey in Mecca.