You asked about how Westerners think about the Chinese names when they are introduced to Chinese. This is definitely a source of confusion for most Westerners, especially if they are not familiar with the Chinese custom of putting the family name first. To an introduction of "My name is Zuo Yu", English Speakers will generally think the person's first (given) name is Zuo and his /her family (surname) is Yu, so you can see that westerners 's who are not experienced in Chinese ways will be completely confused or in error about the name. So they are likely to address the person as"Zuo" (thinking it's a first name) or as Mr. Yu, both of which are incorrect.
To eliminate this problem , I always encourage learners to either
1. give an English first name plus family name OR
2. say" My name is Zuo Yu. You can call me my first name, Yu." OR
3. Reverse the name order when introducing Chinese to Westerners, as " My name is Yu Zuo. " But it would still be a good idea to add, " Yu is my first name " because Westerners cannot usually tell by the sound of a Chinese name which one is a family name and which one a given name (this is usually much easier in English
Hope this helps.
P.S. One additional note: Please be aware that not all Westerners use the other GIVEN NAME + FAMILY NAME order
for introductions. At least some European countries, Hungary for one, use the same order as Chinese. So it is always good to help the listener understand clearly by indicating which name is your GIVEN name.