people in the US have different names, it's true.
A lot of people who come here and have difficult to pronounce names, pick nicknames for themselves. For example, Lucy is not my real name. But ever since I got to the US, I told everyone to call me Lucy, at work, at the doctor's, university, etc. - I am Lucy pretty much officially. My legal name remains my legal name, and it appears on all of my legal documents. Same with my hubby - he has a long unpronounceable Polish name, so he has a nickname. But he is very proud of his name and will never change it.
I have just applied for my citizenship, and I did comtemplate changing my name to Lucy officially. I decided to stick with my real name legally, and continue being Lucy for everything and everybody else.
And I know lots and lots of foreigners (now US citizens or LPRs) who haven't changed their names and never will.

Then, I know some that did. It's ok either way.
If you want to change it before your citizenship, you have to go through court, pay a fee, etc. Then you have to change it on your GC, apply with USCIS, pay a fee again, wait for upto a year....
If you want to wait till your citizenship application, it's done automatically - you just write your new name on the application where they ask you if you want to change your name.
