to that point, wouldn't Spanish people be considered European, and aren't Europeans considered Anglo-saxen white?
You are confused, and you would benefit from studying some more history. The Anglos and the Saxons were germanic tribes that conquered the british isles. Angles - Angleland - England - get it? All of Europe is not considered Anglo-Saxons. Just some of the people in the British Isles, specifically England.
What are known as Hispanics or Latinos in the Americas are of a completely mixed ancestry. The Native Americans in the Islands of the carribean/Mexico/Central America/South America interbred with the European conquerors (Spanish mainly and some Portuguese) and the African slaves that the Europeans brought with them. The level of miscegenation depends on the area.
Now on the the original question. What makes a white person? All of these divisions are based on appearance and typical physical characteristics that are generally considered "white." Since the white race is obviously the fairest race, it takes a low percentage of other genes to drastically darken the skin. Thus affecting appearance. To make it simple if you look at a person and think they are white then... they are white. It has everything to do with appearance. Think about the fact that most black people in the U.S. are a fairly large percentage European in ancestry.
People might want to claim to be a apart of other races because of advantages or simply because they identify with it. For example, being Native American. There are pretty much no full blooded Native Americans around in the U.S. anymore. This just goes to show how arbitrary all of this is.