According to my understanding, "surname" and "last name" are under the same concept -- the "family name" based on the father's side (in general) of the family tree. It is just that the North Americans prefer to say "last name" than "surname". For the Spanish, Portuguese and Latin Americans, they generally have two surnames: surname from the father's side (o apelido do pai) and surname from the mother's side (o apelido da mãe). However, since the majority of culture in the world considers solely the family name from the father's side, people generally refer to their father's surname when speaking with the others. In terms of the people who are under the influence of ancient Chinese culture (China, Taiwan, Korea, Japan, etc.), their tendency is to ALWAYS place the surname at the first place, should it be written in their original language (Reason: to respect the own family name). With the influence of Western culture, however, people now usually follow the Western style if a Christian name is assigned to the person. For instance (your 2nd example), full name in origin: LIU Chen Mei used with a Christian name: Ana Liu (For this case, one may consider that "Ana" is her first name, while "Chen Mei" is her second name, as she possibly prefers to be called "Ana" -- unless you are able to speak in her language.) Thus, when you are asked to inform the "surname" or "last name", you are supposed to provide the surname from the father's side (o apelido do pai); but naturally, if you rather prefer to provide the surname from the mother's side (o apelido da mãe) and you feel comfortable to keep track with that surname, it is surely a freedom of your choice. On the other hand, you may also use both surnames as a combined one by linking them through a hyphen, i.e., '-'. === I wish you a good day.／Desejo que tenhas um bom dia.

The English words refer to names used in the naming system in use in English culture. When you have other systems, it does get confusing! 'Last name' is the same as 'Surname', and doesn't actually have anything to do with where you put it when you write it. That's why surname is a better word to use. So in your example, Liu is both surname and last name, even if you write it first! Yes, it's confusing! One option for Iberian name systems, is to use both family names as the surname, and if necessary / possible, hyphenate them.

It can be confusing. Most commonly, in the United States anyway, because most people have a total of 3 names, 2 given names - their first name and their middle name - and one surname, the father's surname - usually called their last name because it comes last here. e.g. Mary Anne Johnson So, many people who move here with 2 surnames / last names, e.g. María Herrera Rodriguez, end up using only the father's surname (Herrera in that example) or a wife her husband's surname since traditionally (though more exceptions now) the wife simply takes her husband's surname in place of her own, and most forms don't have spaces for more than one last name. And countries where the surname comes first, switch the order or they end up being called wrong. And a surname is the family name, the "last name" here. I've never heard of surname meaning second name unless the second name is the last name.